List of theorems referencing the symbol DOT_INV_IM
(212) Theorem 172: [Domains and ranges of inverse] domain(G) = (G •INV_IM R) + (G •INV_IM (range(G) - R))
(213) Theorem 173: [Range-decomposition of a map domain] ((S * R = 0) & Svm(G)) •imp (((G •INV_IM S) * (G •INV_IM R)) = 0)
(214) Theorem 174: [Inverse image of a range element] (Y in range(G)) •imp ((G •INV_IM {Y}) /= 0)
(215) Theorem 175: [Range of a restricted inverse map] G •INV_IM R = range(inv(G) •ON R)
(216) Theorem 176: [Single-valued inverse image as a setformer] Svm(inv(G)) •imp (G •INV_IM R = {inv(G)~[w]: w in (R * range(G))})
(217) Theorem 177: [Inverse image of a range element] (one_1_map(G) & (Y in range(G))) •imp (G •INV_IM {Y} = {inv(G)~[Y]})
(218) Theorem 178: [Range of a map restricted to an inverse image] range(G •ON (G •INV_IM R)) incs (range(G) * R)
(219) Theorem 179: [Range of a single-valued map restricted to an inverse image] Svm(G) •imp (range(G •ON (G •INV_IM A)) = range(G) * A)
(220) Theorem 180: [Inverse images by a single-valued map] Svm(G) •imp ((G •ON (G •INV_IM A)) •INV_IM B = (G •INV_IM A) * (G •INV_IM B))
(221) Theorem 181: [Monotonicity of inverse image] (B •incin A) •imp ((G •INV_IM B) •incin (G •INV_IM A))
(520) Theorem sigma_theory6: [Rearrangement-of-sums theorem] (Finite(F) & Svm(F) & Svm(G) & (domain(F) = domain(G)) & (range(F) •incin s)) •imp (sigma_thryvar(F) = sigma_thryvar({[y,sigma_thryvar(F •ON (G •INV_IM {y}))]: y in range(G)}))
(900) Theorem 745: (R_SIGMA(0) = R_0) & (FORALL x in OM | (cdr(x) in Re) •imp (R_SIGMA({x}) = cdr(x))) & (FORALL f in OM | (Finite(f) & (range(f) •incin Re)) •imp (R_SIGMA(f) in Re)) & (FORALL f in OM, c in f | (Finite(f) & (range(f) •incin Re)) •imp (R_SIGMA(f) = R_SIGMA(f - {c}) •R_PLUS cdr(c))) & (FORALL f in OM | (Finite(f) & Is_map(f) & (range(f) •incin Re)) •imp (FORALL t | R_SIGMA(f) = R_SIGMA(f •ON (domain(f) * t)) •R_PLUS R_SIGMA(f •ON (domain(f) - t)))) & (FORALL f in OM, g | (Finite(f) & Svm(f) & Svm(g) & (domain(f) = domain(g)) & (range(f) •incin Re)) •imp (R_SIGMA(f) = R_SIGMA({[y,R_SIGMA(f •ON (g •INV_IM {y}))]: y in range(g)}))) & (FORALL f in OM, g | (Finite(f) & Svm(f) & one_1_map(g) & (domain(f) = domain(g)) & (range(f) •incin Re)) •imp (R_SIGMA(f) = R_SIGMA({[y,f~[inv(g)~[y]]]: y in range(g)})))
(901) Theorem 746: (R_SIGMA(0) = R_0) & ((cdr(X) in Re) •imp (R_SIGMA({X}) = cdr(X))) & ((Finite(F) & (range(F) •incin Re)) •imp (R_SIGMA(F) in Re)) & ((Finite(F) & (range(F) •incin Re)) •imp (R_SIGMA(F) = R_SIGMA(F - {c}) •R_PLUS cdr(c))) & ((Finite(F) & Is_map(F) & (range(F) •incin Re)) •imp R_SIGMA(F) = R_SIGMA(F •ON (domain(F) * T)) •R_PLUS R_SIGMA(F •ON (domain(F) - T))) & ((Finite(F) & Svm(F) & Svm(G) & (domain(F) = domain(G)) & (range(F) •incin Re)) •imp (R_SIGMA(F) = R_SIGMA({[y,R_SIGMA(F •ON (G •INV_IM {y}))]: y in range(G)}))) & ((Finite(F) & Svm(F) & one_1_map(G) & (domain(F) = domain(G)) & (range(F) •incin Re)) •imp (R_SIGMA(F) = R_SIGMA({[y,F~[inv(G)~[y]]]: y in range(G)})))