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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