Theorem kqfval 21526

Description: Value of the function appearing in df-kq 21497. (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

Step	Hyp	Ref	Expression
1		id 22	. 2 |- ( A e. X -> A e. X )
2		rabexg 4812	. 2 |- ( J e. V -> { y e. J | A e. y } e. _V )
3		eleq1 2689	. . . 4 |- ( x = A -> ( x e. y <-> A e. y ) )
4	3	rabbidv 3189	. . 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 } )
6	4, 5	fvmptg 6280	. 2 |- ( ( A e. X /\ { y e. J | A e. y } e. _V ) -> ( F ` A ) = { y e. J | A e. y } )
7	1, 2, 6	syl2anr 495	1 |- ( ( J e. V /\ A e. X ) -> ( F ` A ) = { y e. J | A e. y } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   |-> cmpt 4729   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896

This theorem is referenced by:  kqfeq  21527  isr0  21540