Theorem foeq3 3670

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 1484	. . 3 |- (A = B -> (ran F = A <-> ran F = B))
2	1	anbi2d 616	. 2 |- (A = B -> ((F Fn C /\ ran F = A) <-> (F Fn C /\ ran F = B)))
3		df-fo 3196	. 2 |- (F:C-onto->A <-> (F Fn C /\ ran F = A))
4		df-fo 3196	. 2 |- (F:C-onto->B <-> (F Fn C /\ ran F = B))
5	2, 3, 4	3bitr4g 555	1 |- (A = B -> (F:C-onto->A <-> F:C-onto->B))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956  ran crn 3171   Fn wfn 3177  -onto->wfo 3180

This theorem is referenced by:  f1oeq3 3686  ffoss 3711  fodomfiOLD 4566  infmap2lem1 7579  ghsubgi 8138  pjfot 9651

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 963  ax-17 971  ax-4 973  ax-5o 975  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-cleq 1469  df-fo 3196

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