Theorem feq1 5408

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 5362	. . 3 |- ( F = G -> ( F Fn A <-> G Fn A ) )
2		rneq 4905	. . . 4 |- ( F = G -> ran F = ran G )
3	2	sseq1d 3222	. . 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 5275	. 2 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
6		df-f 5275	. 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 1373   C_ wss 3166   ran crn 4676   Fn wfn 5266   -->wf 5267

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-opab 4106  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-fun 5273  df-fn 5274  df-f 5275

This theorem is referenced by:  feq1d  5412  feq1i  5418  f00  5467  f0bi  5468  f0dom0  5469  fconstg  5472  f1eq1  5476  fconst2g  5799  tfrcllemsucfn  6439  tfrcllemsucaccv  6440  tfrcllembxssdm  6442  tfrcllembfn  6443  tfrcllemex  6446  tfrcllemaccex  6447  tfrcllemres  6448  tfrcl  6450  elmapg  6748  ac6sfi  6995  updjud  7184  finomni  7242  exmidomni  7244  mkvprop  7260  1fv  10261  seqf1oglem2  10665  seqf1og  10666  iswrd  10996  isgrpinv  13386  isghm  13579  upxp  14744  txcn  14747  plyf  15209  dceqnconst  15999  dcapnconst  16000