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Theorem isoeq1 5354
Description: Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
Assertion
Ref Expression
isoeq1 (𝐻 = 𝐺 → (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ 𝐺 Isom 𝑅, 𝑆 (A, B)))

Proof of Theorem isoeq1
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1oeq1 5030 . . 3 (𝐻 = 𝐺 → (𝐻:A1-1-ontoB𝐺:A1-1-ontoB))
2 fveq1 5090 . . . . . 6 (𝐻 = 𝐺 → (𝐻x) = (𝐺x))
3 fveq1 5090 . . . . . 6 (𝐻 = 𝐺 → (𝐻y) = (𝐺y))
42, 3breq12d 3740 . . . . 5 (𝐻 = 𝐺 → ((𝐻x)𝑆(𝐻y) ↔ (𝐺x)𝑆(𝐺y)))
54bibi2d 221 . . . 4 (𝐻 = 𝐺 → ((x𝑅y ↔ (𝐻x)𝑆(𝐻y)) ↔ (x𝑅y ↔ (𝐺x)𝑆(𝐺y))))
652ralbidv 2317 . . 3 (𝐻 = 𝐺 → (x A y A (x𝑅y ↔ (𝐻x)𝑆(𝐻y)) ↔ x A y A (x𝑅y ↔ (𝐺x)𝑆(𝐺y))))
71, 6anbi12d 442 . 2 (𝐻 = 𝐺 → ((𝐻:A1-1-ontoB x A y A (x𝑅y ↔ (𝐻x)𝑆(𝐻y))) ↔ (𝐺:A1-1-ontoB x A y A (x𝑅y ↔ (𝐺x)𝑆(𝐺y)))))
8 df-isom 4826 . 2 (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ (𝐻:A1-1-ontoB x A y A (x𝑅y ↔ (𝐻x)𝑆(𝐻y))))
9 df-isom 4826 . 2 (𝐺 Isom 𝑅, 𝑆 (A, B) ↔ (𝐺:A1-1-ontoB x A y A (x𝑅y ↔ (𝐺x)𝑆(𝐺y))))
107, 8, 93bitr4g 212 1 (𝐻 = 𝐺 → (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ 𝐺 Isom 𝑅, 𝑆 (A, B)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1223  wral 2275   class class class wbr 3727  1-1-ontowf1o 4816  cfv 4817   Isom wiso 4818
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 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995
This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1226  df-nf 1323  df-sb 1619  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-ral 2280  df-rex 2281  df-v 2528  df-un 2890  df-in 2892  df-ss 2899  df-sn 3345  df-pr 3346  df-op 3348  df-uni 3544  df-br 3728  df-opab 3782  df-rel 4267  df-cnv 4268  df-co 4269  df-dm 4270  df-rn 4271  df-iota 4782  df-fun 4819  df-fn 4820  df-f 4821  df-f1 4822  df-fo 4823  df-f1o 4824  df-fv 4825  df-isom 4826
This theorem is referenced by:  isores1  5367
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