Theorem foeq1 5586

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 5444	. . 3 |- ( F = G -> ( F Fn A <-> G Fn A ) )
2		rneq 4984	. . . 4 |- ( F = G -> ran F = ran G )
3	2	eqeq1d 2241	. . 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 5358	. 2 |- ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) )
6		df-fo 5358	. 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 1398   ran crn 4750   Fn wfn 5347   -onto->wfo 5350

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-fun 5354  df-fn 5355  df-fo 5358

This theorem is referenced by:  f1oeq1  5602  foeq123d  5607  resdif  5636  dif1en  7136  0ct  7398  ctmlemr  7399  ctm  7400  ctssdclemn0  7401  ctssdclemr  7403  ctssdc  7404  enumct  7406  omct  7408  ctssexmid  7441  exmidfodomrlemim  7504  nninfct  12737  ennnfonelemim  13175  ctinfomlemom  13178  ctinfom  13179  ctinf  13181  qnnen  13182  enctlem  13183  ctiunct  13191  omctfn  13194  ssomct  13196  mndfo  13652  znzrhfo  14796  subctctexmid  16774  domomsubct  16775