Theorem foeq3 2786

Description: Equality theorem for onto functions.

Assertion

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

Proof of Theorem foeq3

Step	Hyp	Ref	Expression
1		cleq2 1110	. . 3 ⊢ (A = B → (ran F = A ↔ ran F = B))
2	1	anbi2d 468	. 2 ⊢ (A = B → ((F Fn C ∧ ran F = A) ↔ (F Fn C ∧ ran F = B)))
3		df-fo 2436	. 2 ⊢ (F:C–onto→A ↔ (F Fn C ∧ ran F = A))
4		df-fo 2436	. 2 ⊢ (F:C–onto→B ↔ (F Fn C ∧ ran F = B))
5	2, 3, 4	3bitr4g 428	1 ⊢ (A = B → (F:C–onto→A ↔ F:C–onto→B))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091  ran crn 2411   Fn wfn 2417  –onto→wfo 2420

This theorem is referenced by:  f1oeq3 2797  ffoss 2820  infmap2lem1 4951

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-cleq 1097  df-fo 2436

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