Theorem isoeq2 5484

Description: Equality theorem for isomorphisms. (Contributed by set.mm contributors, 17-May-2004.)

Assertion

Ref	Expression
isoeq2	⊢ (R = T → (H Isom R, S (A, B) ↔ H Isom T, S (A, B)))

Proof of Theorem isoeq2

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq 4642	. . . . 5 ⊢ (R = T → (xRy ↔ xTy))
2	1	bibi1d 310	. . . 4 ⊢ (R = T → ((xRy ↔ (H ‘x)S(H ‘y)) ↔ (xTy ↔ (H ‘x)S(H ‘y))))
3	2	2ralbidv 2657	. . 3 ⊢ (R = T → (∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y)) ↔ ∀x ∈ A ∀y ∈ A (xTy ↔ (H ‘x)S(H ‘y))))
4	3	anbi2d 684	. 2 ⊢ (R = 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 (xTy ↔ (H ‘x)S(H ‘y)))))
5		df-iso 4797	. 2 ⊢ (H Isom R, S (A, B) ↔ (H:A–1-1-onto→B ∧ ∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y))))
6		df-iso 4797	. 2 ⊢ (H Isom T, S (A, B) ↔ (H:A–1-1-onto→B ∧ ∀x ∈ A ∀y ∈ A (xTy ↔ (H ‘x)S(H ‘y))))
7	4, 5, 6	3bitr4g 279	1 ⊢ (R = T → (H Isom R, S (A, B) ↔ H Isom T, S (A, B)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642  ∀wral 2615   class class class wbr 4640  –1-1-onto→wf1o 4781   ‘cfv 4782   Isom wiso 4783

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-cleq 2346  df-clel 2349  df-ral 2620  df-br 4641  df-iso 4797

This theorem is used by: (None)