Theorem foeq1 5476

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 5346	. . 3 |- ( F = G -> ( F Fn A <-> G Fn A ) )
2		rneq 4893	. . . 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 5264	. 2 |- ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) )
6		df-fo 5264	. 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 4664   Fn wfn 5253   -onto->wfo 5256

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 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-fun 5260  df-fn 5261  df-fo 5264

This theorem is referenced by:  f1oeq1  5492  foeq123d  5497  resdif  5526  dif1en  6940  0ct  7173  ctmlemr  7174  ctm  7175  ctssdclemn0  7176  ctssdclemr  7178  ctssdc  7179  enumct  7181  omct  7183  ctssexmid  7216  exmidfodomrlemim  7268  nninfct  12208  ennnfonelemim  12641  ctinfomlemom  12644  ctinfom  12645  ctinf  12647  qnnen  12648  enctlem  12649  ctiunct  12657  omctfn  12660  ssomct  12662  mndfo  13080  znzrhfo  14204  subctctexmid  15645