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Theorem fvres 5294
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 2618 . . . . 5  |-  x  e. 
_V
21brres 4689 . . . 4  |-  ( A ( F  |`  B ) x  <->  ( A F x  /\  A  e.  B ) )
32rbaib 866 . . 3  |-  ( A  e.  B  ->  ( A ( F  |`  B ) x  <->  A F x ) )
43iotabidv 4969 . 2  |-  ( A  e.  B  ->  ( iota x A ( F  |`  B ) x )  =  ( iota x A F x ) )
5 df-fv 4991 . 2  |-  ( ( F  |`  B ) `  A )  =  ( iota x A ( F  |`  B )
x )
6 df-fv 4991 . 2  |-  ( F `
 A )  =  ( iota x A F x )
74, 5, 63eqtr4g 2142 1  |-  ( A  e.  B  ->  (
( F  |`  B ) `
 A )  =  ( F `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1287    e. wcel 1436   class class class wbr 3822    |` cres 4415   iotacio 4946   ` cfv 4983
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-br 3823  df-opab 3877  df-xp 4419  df-res 4425  df-iota 4948  df-fv 4991
This theorem is referenced by:  funssfv  5295  feqresmpt  5323  fvreseq  5368  respreima  5392  ffvresb  5426  fnressn  5448  fressnfv  5449  fvresi  5455  fvunsng  5456  fvsnun1  5459  fvsnun2  5460  fsnunfv  5462  funfvima  5489  isoresbr  5551  isores3  5557  isoini2  5561  ovres  5743  ofres  5828  offres  5865  fo1stresm  5891  fo2ndresm  5892  fo2ndf  5951  f1o2ndf1  5952  smores  6013  smores2  6015  tfrlem1  6029  rdgival  6103  frec0g  6118  freccllem  6123  frecsuclem  6127  frecrdg  6129  djulclr  6688  djurclr  6689  updjudhcoinlf  6718  updjudhcoinrg  6719  updjud  6720  finomni  6743  exmidfodomrlemrALT  6776  addpiord  6822  mulpiord  6823  fseq1p1m1  9441  iseqfeq2  9807  iseqcoll  10147  shftidt  10167  climres  10589  fisumss  10675  eucialgcvga  10946  eucialg  10947  djucllem  11169
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