Theorem foeq1 5434

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 5304	. . 3 |- ( F = G -> ( F Fn A <-> G Fn A ) )
2		rneq 4854	. . . 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 5222	. 2 |- ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) )
6		df-fo 5222	. 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 4627   Fn wfn 5211   -onto->wfo 5214

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 4004  df-opab 4065  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-fun 5218  df-fn 5219  df-fo 5222

This theorem is referenced by:  f1oeq1  5449  foeq123d  5454  resdif  5483  dif1en  6878  0ct  7105  ctmlemr  7106  ctm  7107  ctssdclemn0  7108  ctssdclemr  7110  ctssdc  7111  enumct  7113  omct  7115  ctssexmid  7147  exmidfodomrlemim  7199  ennnfonelemim  12424  ctinfomlemom  12427  ctinfom  12428  ctinf  12430  qnnen  12431  enctlem  12432  ctiunct  12440  omctfn  12443  ssomct  12445  mndfo  12839  subctctexmid  14720