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Theorem fvtresfn 26698
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 5177 . 2  |-  ( X  e.  B  ->  ( X  |`  V )  e. 
_V )
2 reseq1 5132 . . 3  |-  ( x  =  X  ->  (
x  |`  V )  =  ( X  |`  V ) )
3 fvtresfn.f . . 3  |-  F  =  ( x  e.  B  |->  ( x  |`  V ) )
42, 3fvmptg 5796 . 2  |-  ( ( X  e.  B  /\  ( X  |`  V )  e.  _V )  -> 
( F `  X
)  =  ( X  |`  V ) )
51, 4mpdan 650 1  |-  ( X  e.  B  ->  ( F `  X )  =  ( X  |`  V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   _Vcvv 2948    e. cmpt 4258    |` cres 4872   ` cfv 5446
This theorem is referenced by:  pwssplit1  27120  pwssplit2  27121  pwssplit3  27122  pwssplit4  27123
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-res 4882  df-iota 5410  df-fun 5448  df-fv 5454
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