Theorem isoeq4 5486

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

Assertion

Ref	Expression
isoeq4	⊢ (A = C → (H Isom R, S (A, B) ↔ H Isom R, S (C, B)))

Proof of Theorem isoeq4

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

Step	Hyp	Ref	Expression
1		f1oeq2 5283	. . 3 ⊢ (A = C → (H:A–1-1-onto→B ↔ H:C–1-1-onto→B))
2		raleq 2808	. . . 4 ⊢ (A = C → (∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y)) ↔ ∀y ∈ C (xRy ↔ (H ‘x)S(H ‘y))))
3	2	raleqbi1dv 2816	. . 3 ⊢ (A = C → (∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y)) ↔ ∀x ∈ C ∀y ∈ C (xRy ↔ (H ‘x)S(H ‘y))))
4	1, 3	anbi12d 691	. 2 ⊢ (A = C → ((H:A–1-1-onto→B ∧ ∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y))) ↔ (H:C–1-1-onto→B ∧ ∀x ∈ C ∀y ∈ C (xRy ↔ (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 R, S (C, B) ↔ (H:C–1-1-onto→B ∧ ∀x ∈ C ∀y ∈ C (xRy ↔ (H ‘x)S(H ‘y))))
7	4, 5, 6	3bitr4g 279	1 ⊢ (A = C → (H Isom R, S (A, B) ↔ H Isom R, S (C, B)))

Syntax hints:   → 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 theorem was proved from 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-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-iso 4797

This theorem is referenced by: (None)