Theorem foeq1 5414

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 5284	. . 3 |- ( F = G -> ( F Fn A <-> G Fn A ) )
2		rneq 4836	. . . 4 |- ( F = G -> ran F = ran G )
3	2	eqeq1d 2179	. . 3 |- ( F = G -> ( ran F = B <-> ran G = B ) )
4	1, 3	anbi12d 470	. 2 |- ( F = G -> ( ( F Fn A /\ ran F = B ) <-> ( G Fn A /\ ran G = B ) ) )
5		df-fo 5202	. 2 |- ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) )
6		df-fo 5202	. 2 |- ( G : A -onto-> B <-> ( G Fn A /\ ran G = B ) )
7	4, 5, 6	3bitr4g 222	1 |- ( F = G -> ( F : A -onto-> B <-> G : A -onto-> B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   ran crn 4610   Fn wfn 5191   -onto->wfo 5194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988  df-opab 4049  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-fun 5198  df-fn 5199  df-fo 5202

This theorem is referenced by:  f1oeq1  5429  foeq123d  5434  resdif  5462  dif1en  6853  0ct  7080  ctmlemr  7081  ctm  7082  ctssdclemn0  7083  ctssdclemr  7085  ctssdc  7086  enumct  7088  omct  7090  ctssexmid  7122  exmidfodomrlemim  7165  ennnfonelemim  12366  ctinfomlemom  12369  ctinfom  12370  ctinf  12372  qnnen  12373  enctlem  12374  ctiunct  12382  omctfn  12385  ssomct  12387  mndfo  12662  subctctexmid  13994