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Theorem isoeq1 6552
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 6114 . . 3 (𝐻 = 𝐺 → (𝐻:𝐴1-1-onto𝐵𝐺:𝐴1-1-onto𝐵))
2 fveq1 6177 . . . . . 6 (𝐻 = 𝐺 → (𝐻𝑥) = (𝐺𝑥))
3 fveq1 6177 . . . . . 6 (𝐻 = 𝐺 → (𝐻𝑦) = (𝐺𝑦))
42, 3breq12d 4657 . . . . 5 (𝐻 = 𝐺 → ((𝐻𝑥)𝑆(𝐻𝑦) ↔ (𝐺𝑥)𝑆(𝐺𝑦)))
54bibi2d 332 . . . 4 (𝐻 = 𝐺 → ((𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ (𝑥𝑅𝑦 ↔ (𝐺𝑥)𝑆(𝐺𝑦))))
652ralbidv 2986 . . 3 (𝐻 = 𝐺 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐺𝑥)𝑆(𝐺𝑦))))
71, 6anbi12d 746 . 2 (𝐻 = 𝐺 → ((𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))) ↔ (𝐺:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐺𝑥)𝑆(𝐺𝑦)))))
8 df-isom 5885 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
9 df-isom 5885 . 2 (𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐺:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐺𝑥)𝑆(𝐺𝑦))))
107, 8, 93bitr4g 303 1 (𝐻 = 𝐺 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wral 2909   class class class wbr 4644  1-1-ontowf1o 5875  cfv 5876   Isom wiso 5877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885
This theorem is referenced by:  isores1  6569  wemoiso  7138  wemoiso2  7139  ordiso  8406  oieu  8429  finnisoeu  8921  iunfictbso  8922  infrenegsup  10991  ltweuz  12743  fz1isolem  13228  isercolllem2  14377  isercoll  14379  dvgt0lem2  23747  efcvx  24184  relogiso  24325  logccv  24390  erdszelem1  31147  erdsze  31158  erdsze2lem2  31160  fzisoeu  39327  fourierdlem36  40123  fourierdlem96  40182  fourierdlem97  40183  fourierdlem98  40184  fourierdlem99  40185  fourierdlem105  40191  fourierdlem106  40192  fourierdlem108  40194  fourierdlem110  40196  fourierdlem112  40198  fourierdlem113  40199  fourierdlem115  40201
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