Theorem foeq3 5268

Description: Equality theorem for onto functions. (Contributed by set.mm contributors, 1-Aug-1994.)

Assertion

Ref	Expression
foeq3	⊢ (A = B → (F:C–onto→A ↔ F:C–onto→B))

Proof of Theorem foeq3

Step	Hyp	Ref	Expression
1		eqeq2 2362	. . 3 ⊢ (A = B → (ran F = A ↔ ran F = B))
2	1	anbi2d 684	. 2 ⊢ (A = B → ((F Fn C ∧ ran F = A) ↔ (F Fn C ∧ ran F = B)))
3		df-fo 4794	. 2 ⊢ (F:C–onto→A ↔ (F Fn C ∧ ran F = A))
4		df-fo 4794	. 2 ⊢ (F:C–onto→B ↔ (F Fn C ∧ ran F = B))
5	2, 3, 4	3bitr4g 279	1 ⊢ (A = B → (F:C–onto→A ↔ F:C–onto→B))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642  ran crn 4774   Fn wfn 4777  –onto→wfo 4780

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-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346  df-fo 4794

This theorem is referenced by:  f1oeq3  5284  resdif  5307  ffoss  5315