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Theorem kqfval 17677
Description: Value of the function appearing in df-kq 17648. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
kqval.2  |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )
Assertion
Ref Expression
kqfval  |-  ( ( J  e.  V  /\  A  e.  X )  ->  ( F `  A
)  =  { y  e.  J  |  A  e.  y } )
Distinct variable groups:    x, y, A    x, J, y    x, X, y    x, V
Allowed substitution hints:    F( x, y)    V( y)

Proof of Theorem kqfval
StepHypRef Expression
1 id 20 . 2  |-  ( A  e.  X  ->  A  e.  X )
2 rabexg 4295 . 2  |-  ( J  e.  V  ->  { y  e.  J  |  A  e.  y }  e.  _V )
3 eleq1 2448 . . . 4  |-  ( x  =  A  ->  (
x  e.  y  <->  A  e.  y ) )
43rabbidv 2892 . . 3  |-  ( x  =  A  ->  { y  e.  J  |  x  e.  y }  =  { y  e.  J  |  A  e.  y } )
5 kqval.2 . . 3  |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )
64, 5fvmptg 5744 . 2  |-  ( ( A  e.  X  /\  { y  e.  J  |  A  e.  y }  e.  _V )  ->  ( F `  A )  =  { y  e.  J  |  A  e.  y } )
71, 2, 6syl2anr 465 1  |-  ( ( J  e.  V  /\  A  e.  X )  ->  ( F `  A
)  =  { y  e.  J  |  A  e.  y } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   {crab 2654   _Vcvv 2900    e. cmpt 4208   ` cfv 5395
This theorem is referenced by:  kqfeq  17678  isr0  17691
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-iota 5359  df-fun 5397  df-fv 5403
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