Theorem foeq1 5102

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

Assertion

Ref	Expression
foeq1	|- ( F = G -> ( F : A -onto-> B <-> G : A -onto-> B ) )

Proof of Theorem foeq1

Step	Hyp	Ref	Expression
1		fneq1 4987	. . 3 |- ( F = G -> ( F Fn A <-> G Fn A ) )
2		rneq 4561	. . . 4 |- ( F = G -> ran F = ran G )
3	2	eqeq1d 2048	. . 3 |- ( F = G -> ( ran F = B <-> ran G = B ) )
4	1, 3	anbi12d 442	. 2 |- ( F = G -> ( ( F Fn A /\ ran F = B ) <-> ( G Fn A /\ ran G = B ) ) )
5		df-fo 4908	. 2 |- ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) )
6		df-fo 4908	. 2 |- ( G : A -onto-> B <-> ( G Fn A /\ ran G = B ) )
7	4, 5, 6	3bitr4g 212	1 |- ( F = G -> ( F : A -onto-> B <-> G : A -onto-> B ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   ran crn 4346   Fn wfn 4897   -onto->wfo 4900

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-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-fun 4904  df-fn 4905  df-fo 4908

This theorem is referenced by:  f1oeq1  5117  foeq123d  5122  resdif  5148  dif1en  6337