Theorem foeq1 5431

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 5301	. . 3 |- ( F = G -> ( F Fn A <-> G Fn A ) )
2		rneq 4851	. . . 4 |- ( F = G -> ran F = ran G )
3	2	eqeq1d 2186	. . 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 5219	. 2 |- ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) )
6		df-fo 5219	. 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 1353   ran crn 4625   Fn wfn 5208   -onto->wfo 5211

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-sn 3598  df-pr 3599  df-op 3601  df-br 4002  df-opab 4063  df-rel 4631  df-cnv 4632  df-co 4633  df-dm 4634  df-rn 4635  df-fun 5215  df-fn 5216  df-fo 5219

This theorem is referenced by:  f1oeq1  5446  foeq123d  5451  resdif  5480  dif1en  6874  0ct  7101  ctmlemr  7102  ctm  7103  ctssdclemn0  7104  ctssdclemr  7106  ctssdc  7107  enumct  7109  omct  7111  ctssexmid  7143  exmidfodomrlemim  7195  ennnfonelemim  12415  ctinfomlemom  12418  ctinfom  12419  ctinf  12421  qnnen  12422  enctlem  12423  ctiunct  12431  omctfn  12434  ssomct  12436  mndfo  12770  subctctexmid  14521