Theorem foeq2 3660

Description: Equality theorem for onto functions.

Assertion

Ref	Expression
foeq2	|- (A = B -> (F:A-onto->C <-> F:B-onto->C))

Proof of Theorem foeq2

Step	Hyp	Ref	Expression
1		fneq2 3575	. . 3 |- (A = B -> (F Fn A <-> F Fn B))
2	1	anbi1d 616	. 2 |- (A = B -> ((F Fn A /\ ran F = C) <-> (F Fn B /\ ran F = C)))
3		df-fo 3191	. 2 |- (F:A-onto->C <-> (F Fn A /\ ran F = C))
4		df-fo 3191	. 2 |- (F:B-onto->C <-> (F Fn B /\ ran F = C))
5	2, 3, 4	3bitr4g 554	1 |- (A = B -> (F:A-onto->C <-> F:B-onto->C))

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

This theorem is referenced by:  f1oeq2 3676  fodomfi 4546  fodomg 4779

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-fn 3188  df-fo 3191

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