Theorem isoeq3 2927

Description: Equality theorem for isomorphisms.

Assertion

Ref	Expression
isoeq3	⊢ (S = T → (H Isom R, S (A, B) ↔ H Isom R, T (A, B)))

Proof of Theorem isoeq3

Step	Hyp	Ref	Expression
1		breq 2064	. . . . . 6 ⊢ (S = T → ((H ‘x)S(H ‘y) ↔ (H ‘x)T(H ‘y)))
2	1	bibi2d 470	. . . . 5 ⊢ (S = T → ((xRy ↔ (H ‘x)S(H ‘y)) ↔ (xRy ↔ (H ‘x)T(H ‘y))))
3	2	biraldv 1219	. . . 4 ⊢ (S = T → (∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y)) ↔ ∀y ∈ A (xRy ↔ (H ‘x)T(H ‘y))))
4	3	biraldv 1219	. . 3 ⊢ (S = T → (∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y)) ↔ ∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)T(H ‘y))))
5	4	anbi2d 468	. 2 ⊢ (S = T → ((H:A–1-1-onto→B ∧ ∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y))) ↔ (H:A–1-1-onto→B ∧ ∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)T(H ‘y)))))
6		df-iso 2439	. 2 ⊢ (H Isom R, S (A, B) ↔ (H:A–1-1-onto→B ∧ ∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y))))
7		df-iso 2439	. 2 ⊢ (H Isom R, T (A, B) ↔ (H:A–1-1-onto→B ∧ ∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)T(H ‘y))))
8	5, 6, 7	3bitr4g 428	1 ⊢ (S = T → (H Isom R, S (A, B) ↔ H Isom R, T (A, B)))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091  ∀wral 1201   class class class wbr 2054  –1-1-onto→wf1o 2421   ‘cfv 2422   Isom wiso 2423

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-cleq 1097  df-clel 1099  df-ral 1205  df-br 2063  df-iso 2439

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