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Theorem isoeq1 5594
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 5257 . . 3 (𝐻 = 𝐺 → (𝐻:𝐴1-1-onto𝐵𝐺:𝐴1-1-onto𝐵))
2 fveq1 5317 . . . . . 6 (𝐻 = 𝐺 → (𝐻𝑥) = (𝐺𝑥))
3 fveq1 5317 . . . . . 6 (𝐻 = 𝐺 → (𝐻𝑦) = (𝐺𝑦))
42, 3breq12d 3864 . . . . 5 (𝐻 = 𝐺 → ((𝐻𝑥)𝑆(𝐻𝑦) ↔ (𝐺𝑥)𝑆(𝐺𝑦)))
54bibi2d 231 . . . 4 (𝐻 = 𝐺 → ((𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ (𝑥𝑅𝑦 ↔ (𝐺𝑥)𝑆(𝐺𝑦))))
652ralbidv 2403 . . 3 (𝐻 = 𝐺 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐺𝑥)𝑆(𝐺𝑦))))
71, 6anbi12d 458 . 2 (𝐻 = 𝐺 → ((𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))) ↔ (𝐺:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐺𝑥)𝑆(𝐺𝑦)))))
8 df-isom 5037 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
9 df-isom 5037 . 2 (𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐺:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐺𝑥)𝑆(𝐺𝑦))))
107, 8, 93bitr4g 222 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 103  wb 104   = wceq 1290  wral 2360   class class class wbr 3851  1-1-ontowf1o 5027  cfv 5028   Isom wiso 5029
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-isom 5037
This theorem is referenced by:  isores1  5607  ordiso  6783  infrenegsupex  9143  zfz1isolem1  10306  zfz1iso  10307
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