Theorem fvres 5230

Description: The value of a restricted function. (Contributed by NM, 2-Aug-1994.)

Assertion

Ref	Expression
fvres	|- ( A e. B -> ( ( F |` B ) ` A ) = ( F ` A ) )

Proof of Theorem fvres

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2605	. . . . 5 |- x e. _V
2	1	brres 4646	. . . 4 |- ( A ( F |` B ) x <-> ( A F x /\ A e. B ) )
3	2	rbaib 864	. . 3 |- ( A e. B -> ( A ( F |` B ) x <-> A F x ) )
4	3	iotabidv 4918	. 2 |- ( A e. B -> ( iota x A ( F |` B ) x ) = ( iota x A F x ) )
5		df-fv 4940	. 2 |- ( ( F |` B ) ` A ) = ( iota x A ( F |` B ) x )
6		df-fv 4940	. 2 |- ( F ` A ) = ( iota x A F x )
7	4, 5, 6	3eqtr4g 2139	1 |- ( A e. B -> ( ( F |` B ) ` A ) = ( F ` A ) )

Syntax hints:   -> wi 4   = wceq 1285   e. wcel 1434   class class class wbr 3793   |` cres 4373   iotacio 4895   ` cfv 4932

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-xp 4377  df-res 4383  df-iota 4897  df-fv 4940

This theorem is referenced by:  funssfv  5231  feqresmpt  5259  fvreseq  5303  respreima  5327  ffvresb  5360  fnressn  5381  fressnfv  5382  fvresi  5388  fvunsng  5389  fvsnun1  5392  fvsnun2  5393  fsnunfv  5395  funfvima  5422  isoresbr  5480  isores3  5486  isoini2  5489  ovres  5671  ofres  5756  offres  5793  fo1stresm  5819  fo2ndresm  5820  fo2ndf  5879  f1o2ndf1  5880  smores  5941  smores2  5943  tfrlem1  5957  rdgival  6031  frec0g  6046  freccllem  6051  frecsuclem  6055  frecrdg  6057  addpiord  6568  mulpiord  6569  fseq1p1m1  9187  iseqfeq2  9545  shftidt  9859  climres  10280  eucialgcvga  10584  eucialg  10585