Theorem foeq1 5341

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 5211	. . 3 |- ( F = G -> ( F Fn A <-> G Fn A ) )
2		rneq 4766	. . . 4 |- ( F = G -> ran F = ran G )
3	2	eqeq1d 2148	. . 3 |- ( F = G -> ( ran F = B <-> ran G = B ) )
4	1, 3	anbi12d 464	. 2 |- ( F = G -> ( ( F Fn A /\ ran F = B ) <-> ( G Fn A /\ ran G = B ) ) )
5		df-fo 5129	. 2 |- ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) )
6		df-fo 5129	. 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 1331   ran crn 4540   Fn wfn 5118   -onto->wfo 5121

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-fun 5125  df-fn 5126  df-fo 5129

This theorem is referenced by:  f1oeq1  5356  foeq123d  5361  resdif  5389  dif1en  6773  0ct  6992  ctmlemr  6993  ctm  6994  ctssdclemn0  6995  ctssdclemr  6997  ctssdc  6998  enumct  7000  omct  7002  ctssexmid  7024  exmidfodomrlemim  7057  ennnfonelemim  11937  ctinfomlemom  11940  ctinfom  11941  ctinf  11943  qnnen  11944  enctlem  11945  ctiunct  11953  subctctexmid  13196