Theorem isoeq5 5487

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

Assertion

Ref	Expression
isoeq5	⊢ (B = C → (H Isom R, S (A, B) ↔ H Isom R, S (A, C)))

Proof of Theorem isoeq5

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

Step	Hyp	Ref	Expression
1		f1oeq3 5284	. . 3 ⊢ (B = C → (H:A–1-1-onto→B ↔ H:A–1-1-onto→C))
2	1	anbi1d 685	. 2 ⊢ (B = C → ((H:A–1-1-onto→B ∧ ∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y))) ↔ (H:A–1-1-onto→C ∧ ∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y)))))
3		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))))
4		df-iso 4797	. 2 ⊢ (H Isom R, S (A, C) ↔ (H:A–1-1-onto→C ∧ ∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y))))
5	2, 3, 4	3bitr4g 279	1 ⊢ (B = C → (H Isom R, S (A, B) ↔ H Isom R, S (A, C)))

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-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-iso 4797

This theorem is used by: (None)