Theorem foeq3 3661

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		eqeq2 1481	. . 3 |- (A = B -> (ran F = A <-> ran F = B))
2	1	anbi2d 615	. 2 |- (A = B -> ((F Fn C /\ ran F = A) <-> (F Fn C /\ ran F = B)))
3		df-fo 3191	. 2 |- (F:C-onto->A <-> (F Fn C /\ ran F = A))
4		df-fo 3191	. 2 |- (F:C-onto->B <-> (F Fn C /\ ran F = B))
5	2, 3, 4	3bitr4g 554	1 |- (A = B -> (F:C-onto->A <-> F:C-onto->B))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 954  ran crn 3166   Fn wfn 3172  -onto->wfo 3175

This theorem is referenced by:  f1oeq3 3677  ffoss 3702  fodomfi 4546  infmap2lem1 7529  ghsubgi 8090  pjfot 9591

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 961  ax-17 969  ax-4 971  ax-5o 973  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-cleq 1467  df-fo 3191

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