Theorem foeq1 5611

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 5469	. . 3 |- ( F = G -> ( F Fn A <-> G Fn A ) )
2		rneq 5009	. . . 4 |- ( F = G -> ran F = ran G )
3	2	eqeq1d 2247	. . 3 |- ( F = G -> ( ran F = B <-> ran G = B ) )
4	1, 3	anbi12d 477	. 2 |- ( F = G -> ( ( F Fn A /\ ran F = B ) <-> ( G Fn A /\ ran G = B ) ) )
5		df-fo 5383	. 2 |- ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) )
6		df-fo 5383	. 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 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   ran crn 4775   Fn wfn 5372   -onto->wfo 5375

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3715  df-pr 3716  df-op 3718  df-br 4131  df-opab 4193  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-fun 5379  df-fn 5380  df-fo 5383

This theorem is used by:  f1oeq1  5627  foeq123d  5632  resdif  5661  mapfoss  6947  dif1en  7183  0ct  7447  ctmlemr  7448  ctm  7449  ctssdclemn0  7450  ctssdclemr  7452  ctssdc  7453  enumct  7455  omct  7457  ctssexmid  7490  exmidfodomrlemim  7553  nninfct  12818  ennnfonelemim  13315  ctinfomlemom  13318  ctinfom  13319  ctinf  13321  qnnen  13322  enctlem  13323  ctiunct  13331  omctfn  13334  ssomct  13336  mndfo  13752  znzrhfo  14983  subctctexmid  17030  domomsubct  17031