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Theorem isoeq1 7337
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 6837 . . 3 (𝐻 = 𝐺 → (𝐻:𝐴1-1-onto𝐵𝐺:𝐴1-1-onto𝐵))
2 fveq1 6906 . . . . . 6 (𝐻 = 𝐺 → (𝐻𝑥) = (𝐺𝑥))
3 fveq1 6906 . . . . . 6 (𝐻 = 𝐺 → (𝐻𝑦) = (𝐺𝑦))
42, 3breq12d 5161 . . . . 5 (𝐻 = 𝐺 → ((𝐻𝑥)𝑆(𝐻𝑦) ↔ (𝐺𝑥)𝑆(𝐺𝑦)))
54bibi2d 342 . . . 4 (𝐻 = 𝐺 → ((𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ (𝑥𝑅𝑦 ↔ (𝐺𝑥)𝑆(𝐺𝑦))))
652ralbidv 3219 . . 3 (𝐻 = 𝐺 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐺𝑥)𝑆(𝐺𝑦))))
71, 6anbi12d 632 . 2 (𝐻 = 𝐺 → ((𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))) ↔ (𝐺:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐺𝑥)𝑆(𝐺𝑦)))))
8 df-isom 6572 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
9 df-isom 6572 . 2 (𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐺:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐺𝑥)𝑆(𝐺𝑦))))
107, 8, 93bitr4g 314 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 206  wa 395   = wceq 1537  wral 3059   class class class wbr 5148  1-1-ontowf1o 6562  cfv 6563   Isom wiso 6564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572
This theorem is referenced by:  isores1  7354  wemoiso  7997  wemoiso2  7998  ordiso  9554  oieu  9577  finnisoeu  10151  iunfictbso  10152  infrenegsup  12249  ltweuz  13999  fz1isolem  14497  isercolllem2  15699  isercoll  15701  dvgt0lem2  26057  efcvx  26508  relogiso  26655  logccv  26720  erdszelem1  35176  erdsze  35187  erdsze2lem2  35189  isoeq145d  43409  fzisoeu  45251  fourierdlem36  46099  fourierdlem96  46158  fourierdlem97  46159  fourierdlem98  46160  fourierdlem99  46161  fourierdlem105  46167  fourierdlem106  46168  fourierdlem108  46170  fourierdlem110  46172  fourierdlem112  46174  fourierdlem113  46175  fourierdlem115  46177  rrx2plordisom  48573
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