Theorem feq1 5462

Description: Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)

Assertion

Ref	Expression
feq1	|- ( F = G -> ( F : A --> B <-> G : A --> B ) )

Proof of Theorem feq1

Step	Hyp	Ref	Expression
1		fneq1 5415	. . 3 |- ( F = G -> ( F Fn A <-> G Fn A ) )
2		rneq 4957	. . . 4 |- ( F = G -> ran F = ran G )
3	2	sseq1d 3254	. . 3 |- ( F = G -> ( ran F C_ B <-> ran G C_ B ) )
4	1, 3	anbi12d 473	. 2 |- ( F = G -> ( ( F Fn A /\ ran F C_ B ) <-> ( G Fn A /\ ran G C_ B ) ) )
5		df-f 5328	. 2 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
6		df-f 5328	. 2 |- ( G : A --> B <-> ( G Fn A /\ ran G C_ B ) )
7	4, 5, 6	3bitr4g 223	1 |- ( F = G -> ( F : A --> B <-> G : A --> B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   C_ wss 3198   ran crn 4724   Fn wfn 5319   -->wf 5320

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-fun 5326  df-fn 5327  df-f 5328

This theorem is referenced by:  feq1d  5466  feq1i  5472  f00  5525  f0bi  5526  f0dom0  5527  fconstg  5530  f1eq1  5534  fconst2g  5864  tfrcllemsucfn  6514  tfrcllemsucaccv  6515  tfrcllembxssdm  6517  tfrcllembfn  6518  tfrcllemex  6521  tfrcllemaccex  6522  tfrcllemres  6523  tfrcl  6525  elmapg  6825  ac6sfi  7080  updjud  7272  finomni  7330  exmidomni  7332  mkvprop  7348  1fv  10364  seqf1oglem2  10772  seqf1og  10773  iswrd  11105  isgrpinv  13627  isghm  13820  upxp  14986  txcn  14989  plyf  15451  griedg0prc  16089  dceqnconst  16600  dcapnconst  16601