Theorem foeq1 5472

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 5342	. . 3 |- ( F = G -> ( F Fn A <-> G Fn A ) )
2		rneq 4889	. . . 4 |- ( F = G -> ran F = ran G )
3	2	eqeq1d 2202	. . 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 5260	. 2 |- ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) )
6		df-fo 5260	. 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 4660   Fn wfn 5249   -onto->wfo 5252

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-fun 5256  df-fn 5257  df-fo 5260

This theorem is referenced by:  f1oeq1  5488  foeq123d  5493  resdif  5522  dif1en  6935  0ct  7166  ctmlemr  7167  ctm  7168  ctssdclemn0  7169  ctssdclemr  7171  ctssdc  7172  enumct  7174  omct  7176  ctssexmid  7209  exmidfodomrlemim  7261  nninfct  12178  ennnfonelemim  12581  ctinfomlemom  12584  ctinfom  12585  ctinf  12587  qnnen  12588  enctlem  12589  ctiunct  12597  omctfn  12600  ssomct  12602  mndfo  13020  znzrhfo  14136  subctctexmid  15491