Theorem foeq1 5544

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  5560  foeq123d  5565  resdif  5594  dif1en  7041  0ct  7274  ctmlemr  7275  ctm  7276  ctssdclemn0  7277  ctssdclemr  7279  ctssdc  7280  enumct  7282  omct  7284  ctssexmid  7317  exmidfodomrlemim  7379  nninfct  12562  ennnfonelemim  12995  ctinfomlemom  12998  ctinfom  12999  ctinf  13001  qnnen  13002  enctlem  13003  ctiunct  13011  omctfn  13014  ssomct  13016  mndfo  13472  znzrhfo  14612  subctctexmid  16366  domomsubct  16367