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Theorem fvres 5198
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.
StepHypRef Expression
1 vex 2560 . . . . 5  |-  x  e. 
_V
21brres 4618 . . . 4  |-  ( A ( F  |`  B ) x  <->  ( A F x  /\  A  e.  B ) )
32rbaib 830 . . 3  |-  ( A  e.  B  ->  ( A ( F  |`  B ) x  <->  A F x ) )
43iotabidv 4888 . 2  |-  ( A  e.  B  ->  ( iota x A ( F  |`  B ) x )  =  ( iota x A F x ) )
5 df-fv 4910 . 2  |-  ( ( F  |`  B ) `  A )  =  ( iota x A ( F  |`  B )
x )
6 df-fv 4910 . 2  |-  ( F `
 A )  =  ( iota x A F x )
74, 5, 63eqtr4g 2097 1  |-  ( A  e.  B  ->  (
( F  |`  B ) `
 A )  =  ( F `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1243    e. wcel 1393   class class class wbr 3764    |` cres 4347   iotacio 4865   ` cfv 4902
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-xp 4351  df-res 4357  df-iota 4867  df-fv 4910
This theorem is referenced by:  funssfv  5199  feqresmpt  5227  fvreseq  5271  respreima  5295  ffvresb  5328  fnressn  5349  fressnfv  5350  fvresi  5356  fvunsng  5357  fvsnun1  5360  fvsnun2  5361  fsnunfv  5363  funfvima  5390  isoresbr  5449  isores3  5455  isoini2  5458  ovres  5640  ofres  5725  offres  5762  fo1stresm  5788  fo2ndresm  5789  fo2ndf  5848  f1o2ndf1  5849  smores  5907  smores2  5909  tfrlem1  5923  rdgival  5969  rdgon  5973  frec0g  5983  frecsuclem1  5987  frecsuclem2  5989  frecrdg  5992  addpiord  6414  mulpiord  6415  fseq1p1m1  8956  iseqfeq2  9229  shftidt  9434  climres  9824
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