Theorem foeq1 5405

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 5275	. . 3 |- ( F = G -> ( F Fn A <-> G Fn A ) )
2		rneq 4830	. . . 4 |- ( F = G -> ran F = ran G )
3	2	eqeq1d 2174	. . 3 |- ( F = G -> ( ran F = B <-> ran G = B ) )
4	1, 3	anbi12d 465	. 2 |- ( F = G -> ( ( F Fn A /\ ran F = B ) <-> ( G Fn A /\ ran G = B ) ) )
5		df-fo 5193	. 2 |- ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) )
6		df-fo 5193	. 2 |- ( G : A -onto-> B <-> ( G Fn A /\ ran G = B ) )
7	4, 5, 6	3bitr4g 222	1 |- ( F = G -> ( F : A -onto-> B <-> G : A -onto-> B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   ran crn 4604   Fn wfn 5182   -onto->wfo 5185

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-v 2727  df-un 3119  df-in 3121  df-ss 3128  df-sn 3581  df-pr 3582  df-op 3584  df-br 3982  df-opab 4043  df-rel 4610  df-cnv 4611  df-co 4612  df-dm 4613  df-rn 4614  df-fun 5189  df-fn 5190  df-fo 5193

This theorem is referenced by:  f1oeq1  5420  foeq123d  5425  resdif  5453  dif1en  6841  0ct  7068  ctmlemr  7069  ctm  7070  ctssdclemn0  7071  ctssdclemr  7073  ctssdc  7074  enumct  7076  omct  7078  ctssexmid  7110  exmidfodomrlemim  7153  ennnfonelemim  12353  ctinfomlemom  12356  ctinfom  12357  ctinf  12359  qnnen  12360  enctlem  12361  ctiunct  12369  omctfn  12372  ssomct  12374  subctctexmid  13841