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Theorem kqfval 17745
Description: Value of the function appearing in df-kq 17716. (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 4345 . 2  |-  ( J  e.  V  ->  { y  e.  J  |  A  e.  y }  e.  _V )
3 eleq1 2495 . . . 4  |-  ( x  =  A  ->  (
x  e.  y  <->  A  e.  y ) )
43rabbidv 2940 . . 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 5796 . 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 1652    e. wcel 1725   {crab 2701   _Vcvv 2948    e. cmpt 4258   ` cfv 5446
This theorem is referenced by:  kqfeq  17746  isr0  17759
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-iota 5410  df-fun 5448  df-fv 5454
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