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Theorem fvtresfn 5932
Description: Functionality of a tuple-restriction function. (Contributed by Stefan O'Rear, 24-Jan-2015.)
Hypothesis
Ref Expression
fvtresfn.f  |-  F  =  ( x  e.  B  |->  ( x  |`  V ) )
Assertion
Ref Expression
fvtresfn  |-  ( X  e.  B  ->  ( F `  X )  =  ( X  |`  V ) )
Distinct variable groups:    x, B    x, V    x, X
Allowed substitution hint:    F( x)

Proof of Theorem fvtresfn
StepHypRef Expression
1 resexg 5304 . 2  |-  ( X  e.  B  ->  ( X  |`  V )  e. 
_V )
2 reseq1 5256 . . 3  |-  ( x  =  X  ->  (
x  |`  V )  =  ( X  |`  V ) )
3 fvtresfn.f . . 3  |-  F  =  ( x  e.  B  |->  ( x  |`  V ) )
42, 3fvmptg 5929 . 2  |-  ( ( X  e.  B  /\  ( X  |`  V )  e.  _V )  -> 
( F `  X
)  =  ( X  |`  V ) )
51, 4mpdan 666 1  |-  ( X  e.  B  ->  ( F `  X )  =  ( X  |`  V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823   _Vcvv 3106    |-> cmpt 4497    |` cres 4990   ` cfv 5570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-res 5000  df-iota 5534  df-fun 5572  df-fv 5578
This theorem is referenced by:  symgfixf1  16661  symgfixfo  16663  pwssplit1  17900  pwssplit2  17901  pwssplit3  17902  eulerpartgbij  28575  pwssplit4  31274
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