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Theorem isoeq1 7220
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 6734 . . 3 (𝐻 = 𝐺 → (𝐻:𝐴1-1-onto𝐵𝐺:𝐴1-1-onto𝐵))
2 fveq1 6803 . . . . . 6 (𝐻 = 𝐺 → (𝐻𝑥) = (𝐺𝑥))
3 fveq1 6803 . . . . . 6 (𝐻 = 𝐺 → (𝐻𝑦) = (𝐺𝑦))
42, 3breq12d 5094 . . . . 5 (𝐻 = 𝐺 → ((𝐻𝑥)𝑆(𝐻𝑦) ↔ (𝐺𝑥)𝑆(𝐺𝑦)))
54bibi2d 343 . . . 4 (𝐻 = 𝐺 → ((𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ (𝑥𝑅𝑦 ↔ (𝐺𝑥)𝑆(𝐺𝑦))))
652ralbidv 3209 . . 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 6467 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
9 df-isom 6467 . 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 205  wa 397   = wceq 1539  wral 3062   class class class wbr 5081  1-1-ontowf1o 6457  cfv 6458   Isom wiso 6459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3063  df-rab 3287  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-opab 5144  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-isom 6467
This theorem is referenced by:  isores1  7237  wemoiso  7848  wemoiso2  7849  ordiso  9319  oieu  9342  finnisoeu  9915  iunfictbso  9916  infrenegsup  12004  ltweuz  13727  fz1isolem  14220  isercolllem2  15422  isercoll  15424  dvgt0lem2  25212  efcvx  25653  relogiso  25798  logccv  25863  erdszelem1  33198  erdsze  33209  erdsze2lem2  33211  fzisoeu  42887  fourierdlem36  43733  fourierdlem96  43792  fourierdlem97  43793  fourierdlem98  43794  fourierdlem99  43795  fourierdlem105  43801  fourierdlem106  43802  fourierdlem108  43804  fourierdlem110  43806  fourierdlem112  43808  fourierdlem113  43809  fourierdlem115  43811  rrx2plordisom  46127
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