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Theorem isoeq1 5384
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 5060 . . 3 (𝐻 = 𝐺 → (𝐻:A1-1-ontoB𝐺:A1-1-ontoB))
2 fveq1 5120 . . . . . 6 (𝐻 = 𝐺 → (𝐻x) = (𝐺x))
3 fveq1 5120 . . . . . 6 (𝐻 = 𝐺 → (𝐻y) = (𝐺y))
42, 3breq12d 3768 . . . . 5 (𝐻 = 𝐺 → ((𝐻x)𝑆(𝐻y) ↔ (𝐺x)𝑆(𝐺y)))
54bibi2d 221 . . . 4 (𝐻 = 𝐺 → ((x𝑅y ↔ (𝐻x)𝑆(𝐻y)) ↔ (x𝑅y ↔ (𝐺x)𝑆(𝐺y))))
652ralbidv 2342 . . 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 4854 . 2 (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ (𝐻:A1-1-ontoB x A y A (x𝑅y ↔ (𝐻x)𝑆(𝐻y))))
9 df-isom 4854 . 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 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-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-isom 4854
This theorem is referenced by:  isores1  5397
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