Theorem setsvalg 12862

Description: Value of the structure replacement function. (Contributed by Mario Carneiro, 30-Apr-2015.)

Assertion

Ref	Expression
setsvalg	|- ( ( S e. V /\ A e. W ) -> ( S sSet A ) = ( ( S |` ( _V \ dom { A } ) ) u. { A } ) )

Proof of Theorem setsvalg

Dummy variables e s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2783	. 2 |- ( S e. V -> S e. _V )
2		elex 2783	. 2 |- ( A e. W -> A e. _V )
3		resexg 4999	. . . 4 |- ( S e. _V -> ( S |` ( _V \ dom { A } ) ) e. _V )
4		snexg 4228	. . . 4 |- ( A e. _V -> { A } e. _V )
5		unexg 4490	. . . 4 |- ( ( ( S |` ( _V \ dom { A } ) ) e. _V /\ { A } e. _V ) -> ( ( S |` ( _V \ dom { A } ) ) u. { A } ) e. _V )
6	3, 4, 5	syl2an 289	. . 3 |- ( ( S e. _V /\ A e. _V ) -> ( ( S |` ( _V \ dom { A } ) ) u. { A } ) e. _V )
7		simpl 109	. . . . . 6 |- ( ( s = S /\ e = A ) -> s = S )
8		simpr 110	. . . . . . . . 9 |- ( ( s = S /\ e = A ) -> e = A )
9	8	sneqd 3646	. . . . . . . 8 |- ( ( s = S /\ e = A ) -> { e } = { A } )
10	9	dmeqd 4880	. . . . . . 7 |- ( ( s = S /\ e = A ) -> dom { e } = dom { A } )
11	10	difeq2d 3291	. . . . . 6 |- ( ( s = S /\ e = A ) -> ( _V \ dom { e } ) = ( _V \ dom { A } ) )
12	7, 11	reseq12d 4960	. . . . 5 |- ( ( s = S /\ e = A ) -> ( s |` ( _V \ dom { e } ) ) = ( S |` ( _V \ dom { A } ) ) )
13	12, 9	uneq12d 3328	. . . 4 |- ( ( s = S /\ e = A ) -> ( ( s |` ( _V \ dom { e } ) ) u. { e } ) = ( ( S |` ( _V \ dom { A } ) ) u. { A } ) )
14		df-sets 12839	. . . 4 |- sSet = ( s e. _V , e e. _V |-> ( ( s |` ( _V \ dom { e } ) ) u. { e } ) )
15	13, 14	ovmpoga 6075	. . 3 |- ( ( S e. _V /\ A e. _V /\ ( ( S |` ( _V \ dom { A } ) ) u. { A } ) e. _V ) -> ( S sSet A ) = ( ( S |` ( _V \ dom { A } ) ) u. { A } ) )
16	6, 15	mpd3an3 1351	. 2 |- ( ( S e. _V /\ A e. _V ) -> ( S sSet A ) = ( ( S |` ( _V \ dom { A } ) ) u. { A } ) )
17	1, 2, 16	syl2an 289	1 |- ( ( S e. V /\ A e. W ) -> ( S sSet A ) = ( ( S |` ( _V \ dom { A } ) ) u. { A } ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   _Vcvv 2772   \ cdif 3163   u. cun 3164   {csn 3633   dom cdm 4675   |` cres 4677  (class class class)co 5944   sSet csts 12830

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-res 4687  df-iota 5232  df-fun 5273  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-sets 12839

This theorem is referenced by:  setsvala  12863  setsfun  12867  setsfun0  12868  setsresg  12870