Theorem isoeq1 7168

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.

Step	Hyp	Ref	Expression
1		f1oeq1 6688	. . 3 ⊢ (𝐻 = 𝐺 → (𝐻:𝐴–1-1-onto→𝐵 ↔ 𝐺:𝐴–1-1-onto→𝐵))
2		fveq1 6755	. . . . . 6 ⊢ (𝐻 = 𝐺 → (𝐻‘𝑥) = (𝐺‘𝑥))
3		fveq1 6755	. . . . . 6 ⊢ (𝐻 = 𝐺 → (𝐻‘𝑦) = (𝐺‘𝑦))
4	2, 3	breq12d 5083	. . . . 5 ⊢ (𝐻 = 𝐺 → ((𝐻‘𝑥)𝑆(𝐻‘𝑦) ↔ (𝐺‘𝑥)𝑆(𝐺‘𝑦)))
5	4	bibi2d 342	. . . 4 ⊢ (𝐻 = 𝐺 → ((𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ (𝑥𝑅𝑦 ↔ (𝐺‘𝑥)𝑆(𝐺‘𝑦))))
6	5	2ralbidv 3122	. . 3 ⊢ (𝐻 = 𝐺 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐺‘𝑥)𝑆(𝐺‘𝑦))))
7	1, 6	anbi12d 630	. 2 ⊢ (𝐻 = 𝐺 → ((𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))) ↔ (𝐺:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐺‘𝑥)𝑆(𝐺‘𝑦)))))
8		df-isom 6427	. 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
9		df-isom 6427	. 2 ⊢ (𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐺:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐺‘𝑥)𝑆(𝐺‘𝑦))))
10	7, 8, 9	3bitr4g 313	1 ⊢ (𝐻 = 𝐺 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539  ∀wral 3063   class class class wbr 5070  –1-1-onto→wf1o 6417  ‘cfv 6418   Isom wiso 6419

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427

This theorem is referenced by:  isores1  7185  wemoiso  7789  wemoiso2  7790  ordiso  9205  oieu  9228  finnisoeu  9800  iunfictbso  9801  infrenegsup  11888  ltweuz  13609  fz1isolem  14103  isercolllem2  15305  isercoll  15307  dvgt0lem2  25072  efcvx  25513  relogiso  25658  logccv  25723  erdszelem1  33053  erdsze  33064  erdsze2lem2  33066  fzisoeu  42729  fourierdlem36  43574  fourierdlem96  43633  fourierdlem97  43634  fourierdlem98  43635  fourierdlem99  43636  fourierdlem105  43642  fourierdlem106  43643  fourierdlem108  43645  fourierdlem110  43647  fourierdlem112  43649  fourierdlem113  43650  fourierdlem115  43652  rrx2plordisom  45957