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Theorem fvtresfn 5874
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 5247 . 2  |-  ( X  e.  B  ->  ( X  |`  V )  e. 
_V )
2 reseq1 5202 . . 3  |-  ( x  =  X  ->  (
x  |`  V )  =  ( X  |`  V ) )
3 fvtresfn.f . . 3  |-  F  =  ( x  e.  B  |->  ( x  |`  V ) )
42, 3fvmptg 5871 . 2  |-  ( ( X  e.  B  /\  ( X  |`  V )  e.  _V )  -> 
( F `  X
)  =  ( X  |`  V ) )
51, 4mpdan 668 1  |-  ( X  e.  B  ->  ( F `  X )  =  ( X  |`  V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   _Vcvv 3068    |-> cmpt 4448    |` cres 4940   ` cfv 5516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-sbc 3285  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-res 4950  df-iota 5479  df-fun 5518  df-fv 5524
This theorem is referenced by:  symgfixfo  16047  pwssplit1  17246  pwssplit2  17247  pwssplit3  17248  pwssplit4  29580
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