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.

Step	Hyp	Ref	Expression
1		f1oeq1 5257	. . 3 ⊢ (𝐻 = 𝐺 → (𝐻:𝐴–1-1-onto→𝐵 ↔ 𝐺:𝐴–1-1-onto→𝐵))
2		fveq1 5317	. . . . . 6 ⊢ (𝐻 = 𝐺 → (𝐻‘𝑥) = (𝐺‘𝑥))
3		fveq1 5317	. . . . . 6 ⊢ (𝐻 = 𝐺 → (𝐻‘𝑦) = (𝐺‘𝑦))
4	2, 3	breq12d 3864	. . . . 5 ⊢ (𝐻 = 𝐺 → ((𝐻‘𝑥)𝑆(𝐻‘𝑦) ↔ (𝐺‘𝑥)𝑆(𝐺‘𝑦)))
5	4	bibi2d 231	. . . 4 ⊢ (𝐻 = 𝐺 → ((𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ (𝑥𝑅𝑦 ↔ (𝐺‘𝑥)𝑆(𝐺‘𝑦))))
6	5	2ralbidv 2403	. . 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 458	. 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 5037	. 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 5037	. 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 222	1 ⊢ (𝐻 = 𝐺 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1290  ∀wral 2360   class class class wbr 3851  –1-1-onto→wf1o 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