Theorem setsvalg 13102

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 2812	. 2 |- ( S e. V -> S e. _V )
2		elex 2812	. 2 |- ( A e. W -> A e. _V )
3		resexg 5051	. . . 4 |- ( S e. _V -> ( S |` ( _V \ dom { A } ) ) e. _V )
4		snexg 4272	. . . 4 |- ( A e. _V -> { A } e. _V )
5		unexg 4538	. . . 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 3680	. . . . . . . 8 |- ( ( s = S /\ e = A ) -> { e } = { A } )
10	9	dmeqd 4931	. . . . . . 7 |- ( ( s = S /\ e = A ) -> dom { e } = dom { A } )
11	10	difeq2d 3323	. . . . . 6 |- ( ( s = S /\ e = A ) -> ( _V \ dom { e } ) = ( _V \ dom { A } ) )
12	7, 11	reseq12d 5012	. . . . 5 |- ( ( s = S /\ e = A ) -> ( s |` ( _V \ dom { e } ) ) = ( S |` ( _V \ dom { A } ) ) )
13	12, 9	uneq12d 3360	. . . 4 |- ( ( s = S /\ e = A ) -> ( ( s |` ( _V \ dom { e } ) ) u. { e } ) = ( ( S |` ( _V \ dom { A } ) ) u. { A } ) )
14		df-sets 13079	. . . 4 |- sSet = ( s e. _V , e e. _V |-> ( ( s |` ( _V \ dom { e } ) ) u. { e } ) )
15	13, 14	ovmpoga 6146	. . 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 1372	. 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 1395   e. wcel 2200   _Vcvv 2800   \ cdif 3195   u. cun 3196   {csn 3667   dom cdm 4723   |` cres 4725  (class class class)co 6013   sSet csts 13070

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-res 4735  df-iota 5284  df-fun 5326  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-sets 13079

This theorem is referenced by:  setsvala  13103  setsfun  13107  setsfun0  13108  setsresg  13110  bassetsnn  13129