Theorem foeq1 5546

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 5409	. . 3 |- ( F = G -> ( F Fn A <-> G Fn A ) )
2		rneq 4951	. . . 4 |- ( F = G -> ran F = ran G )
3	2	eqeq1d 2238	. . 3 |- ( F = G -> ( ran F = B <-> ran G = B ) )
4	1, 3	anbi12d 473	. 2 |- ( F = G -> ( ( F Fn A /\ ran F = B ) <-> ( G Fn A /\ ran G = B ) ) )
5		df-fo 5324	. 2 |- ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) )
6		df-fo 5324	. 2 |- ( G : A -onto-> B <-> ( G Fn A /\ ran G = B ) )
7	4, 5, 6	3bitr4g 223	1 |- ( F = G -> ( F : A -onto-> B <-> G : A -onto-> B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   ran crn 4720   Fn wfn 5313   -onto->wfo 5316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-fun 5320  df-fn 5321  df-fo 5324

This theorem is referenced by:  f1oeq1  5562  foeq123d  5567  resdif  5596  dif1en  7049  0ct  7285  ctmlemr  7286  ctm  7287  ctssdclemn0  7288  ctssdclemr  7290  ctssdc  7291  enumct  7293  omct  7295  ctssexmid  7328  exmidfodomrlemim  7390  nninfct  12578  ennnfonelemim  13011  ctinfomlemom  13014  ctinfom  13015  ctinf  13017  qnnen  13018  enctlem  13019  ctiunct  13027  omctfn  13030  ssomct  13032  mndfo  13488  znzrhfo  14628  subctctexmid  16453  domomsubct  16454