Theorem setsvalg 13242

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 2825	. 2 |- ( S e. V -> S e. _V )
2		elex 2825	. 2 |- ( A e. W -> A e. _V )
3		resexg 5078	. . . 4 |- ( S e. _V -> ( S |` ( _V \ dom { A } ) ) e. _V )
4		snexg 4297	. . . 4 |- ( A e. _V -> { A } e. _V )
5		unexg 4564	. . . 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 3702	. . . . . . . 8 |- ( ( s = S /\ e = A ) -> { e } = { A } )
10	9	dmeqd 4958	. . . . . . 7 |- ( ( s = S /\ e = A ) -> dom { e } = dom { A } )
11	10	difeq2d 3337	. . . . . 6 |- ( ( s = S /\ e = A ) -> ( _V \ dom { e } ) = ( _V \ dom { A } ) )
12	7, 11	reseq12d 5039	. . . . 5 |- ( ( s = S /\ e = A ) -> ( s |` ( _V \ dom { e } ) ) = ( S |` ( _V \ dom { A } ) ) )
13	12, 9	uneq12d 3374	. . . 4 |- ( ( s = S /\ e = A ) -> ( ( s |` ( _V \ dom { e } ) ) u. { e } ) = ( ( S |` ( _V \ dom { A } ) ) u. { A } ) )
14		df-sets 13219	. . . 4 |- sSet = ( s e. _V , e e. _V |-> ( ( s |` ( _V \ dom { e } ) ) u. { e } ) )
15	13, 14	ovmpoga 6183	. . 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 1375	. 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 1398   e. wcel 2203   _Vcvv 2813   \ cdif 3208   u. cun 3209   {csn 3689   dom cdm 4749   |` cres 4751  (class class class)co 6050   sSet csts 13210

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-res 4761  df-iota 5312  df-fun 5354  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-sets 13219

This theorem is referenced by:  setsvala  13243  setsfun  13247  setsfun0  13248  setsresg  13250  bassetsnn  13269