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Theorem isoeq4 5387
Description: Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
Assertion
Ref Expression
isoeq4 (A = 𝐶 → (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ 𝐻 Isom 𝑅, 𝑆 (𝐶, B)))

Proof of Theorem isoeq4
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1oeq2 5061 . . 3 (A = 𝐶 → (𝐻:A1-1-ontoB𝐻:𝐶1-1-ontoB))
2 raleq 2499 . . . 4 (A = 𝐶 → (y A (x𝑅y ↔ (𝐻x)𝑆(𝐻y)) ↔ y 𝐶 (x𝑅y ↔ (𝐻x)𝑆(𝐻y))))
32raleqbi1dv 2507 . . 3 (A = 𝐶 → (x A y A (x𝑅y ↔ (𝐻x)𝑆(𝐻y)) ↔ x 𝐶 y 𝐶 (x𝑅y ↔ (𝐻x)𝑆(𝐻y))))
41, 3anbi12d 442 . 2 (A = 𝐶 → ((𝐻:A1-1-ontoB x A y A (x𝑅y ↔ (𝐻x)𝑆(𝐻y))) ↔ (𝐻:𝐶1-1-ontoB x 𝐶 y 𝐶 (x𝑅y ↔ (𝐻x)𝑆(𝐻y)))))
5 df-isom 4854 . 2 (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ (𝐻:A1-1-ontoB x A y A (x𝑅y ↔ (𝐻x)𝑆(𝐻y))))
6 df-isom 4854 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐶, B) ↔ (𝐻:𝐶1-1-ontoB x 𝐶 y 𝐶 (x𝑅y ↔ (𝐻x)𝑆(𝐻y))))
74, 5, 63bitr4g 212 1 (A = 𝐶 → (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ 𝐻 Isom 𝑅, 𝑆 (𝐶, B)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  wral 2300   class class class wbr 3755  1-1-ontowf1o 4844  cfv 4845   Isom wiso 4846
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-isom 4854
This theorem is referenced by: (None)
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