Theorem foeq2 5046

Description: Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)

Assertion

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

Proof of Theorem foeq2

Step	Hyp	Ref	Expression
1		fneq2 4931	. . 3 |- (A = B -> ( F Fn A <-> F Fn B))
2	1	anbi1d 438	. 2 |- (A = B -> (( F Fn A /\ ran F = C) <-> ( F Fn B /\ ran F = C)))
3		df-fo 4851	. 2 |- ( F :A -onto-> C <-> ( F Fn A /\ ran F = C))
4		df-fo 4851	. 2 |- ( F :B -onto-> C <-> ( F Fn B /\ ran F = C))
5	2, 3, 4	3bitr4g 212	1 |- (A = B -> ( F :A -onto-> C <-> F :B -onto-> C))

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1242   ran crn 4289   Fn wfn 4840   -onto->wfo 4843

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-4 1397  ax-17 1416  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-cleq 2030  df-fn 4848  df-fo 4851

This theorem is referenced by:  f1oeq2  5061  foeq123d  5065  tposfo  5827