Theorem isoeq1 5780

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 5431	. . 3 ⊢ (𝐻 = 𝐺 → (𝐻:𝐴–1-1-onto→𝐵 ↔ 𝐺:𝐴–1-1-onto→𝐵))
2		fveq1 5495	. . . . . 6 ⊢ (𝐻 = 𝐺 → (𝐻‘𝑥) = (𝐺‘𝑥))
3		fveq1 5495	. . . . . 6 ⊢ (𝐻 = 𝐺 → (𝐻‘𝑦) = (𝐺‘𝑦))
4	2, 3	breq12d 4002	. . . . 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 2494	. . 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 470	. 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 5207	. 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 5207	. 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 1348  ∀wral 2448   class class class wbr 3989  –1-1-onto→wf1o 5197  ‘cfv 5198   Isom wiso 5199

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207

This theorem is referenced by:  isores1  5793  ordiso  7013  infrenegsupex  9553  zfz1isolem1  10775  zfz1iso  10776  infxrnegsupex  11226  relogiso  13588