Theorem foeq1 5479

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 5347	. . 3 |- ( F = G -> ( F Fn A <-> G Fn A ) )
2		rneq 4894	. . . 4 |- ( F = G -> ran F = ran G )
3	2	eqeq1d 2205	. . 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 5265	. 2 |- ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) )
6		df-fo 5265	. 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 1364   ran crn 4665   Fn wfn 5254   -onto->wfo 5257

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-opab 4096  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-fun 5261  df-fn 5262  df-fo 5265

This theorem is referenced by:  f1oeq1  5495  foeq123d  5500  resdif  5529  dif1en  6949  0ct  7182  ctmlemr  7183  ctm  7184  ctssdclemn0  7185  ctssdclemr  7187  ctssdc  7188  enumct  7190  omct  7192  ctssexmid  7225  exmidfodomrlemim  7280  nninfct  12233  ennnfonelemim  12666  ctinfomlemom  12669  ctinfom  12670  ctinf  12672  qnnen  12673  enctlem  12674  ctiunct  12682  omctfn  12685  ssomct  12687  mndfo  13141  znzrhfo  14280  subctctexmid  15731  domomsubct  15732