Theorem foeq1 5552

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 5415	. . 3 |- ( F = G -> ( F Fn A <-> G Fn A ) )
2		rneq 4957	. . . 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 5330	. 2 |- ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) )
6		df-fo 5330	. 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 4724   Fn wfn 5319   -onto->wfo 5322

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 2802  df-un 3202  df-in 3204  df-ss 3211  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-fun 5326  df-fn 5327  df-fo 5330

This theorem is referenced by:  f1oeq1  5568  foeq123d  5573  resdif  5602  dif1en  7061  0ct  7297  ctmlemr  7298  ctm  7299  ctssdclemn0  7300  ctssdclemr  7302  ctssdc  7303  enumct  7305  omct  7307  ctssexmid  7340  exmidfodomrlemim  7402  nninfct  12602  ennnfonelemim  13035  ctinfomlemom  13038  ctinfom  13039  ctinf  13041  qnnen  13042  enctlem  13043  ctiunct  13051  omctfn  13054  ssomct  13056  mndfo  13512  znzrhfo  14652  subctctexmid  16537  domomsubct  16538