Theorem setsvalg 15887

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 3212	. 2 |- ( S e. V -> S e. _V )
2		elex 3212	. 2 |- ( A e. W -> A e. _V )
3		resexg 5442	. . . . 5 |- ( S e. _V -> ( S |` ( _V \ dom { A } ) ) e. _V )
4	3	adantr 481	. . . 4 |- ( ( S e. _V /\ A e. _V ) -> ( S |` ( _V \ dom { A } ) ) e. _V )
5		snex 4908	. . . 4 |- { A } e. _V
6		unexg 6959	. . . 4 |- ( ( ( S |` ( _V \ dom { A } ) ) e. _V /\ { A } e. _V ) -> ( ( S |` ( _V \ dom { A } ) ) u. { A } ) e. _V )
7	4, 5, 6	sylancl 694	. . 3 |- ( ( S e. _V /\ A e. _V ) -> ( ( S |` ( _V \ dom { A } ) ) u. { A } ) e. _V )
8		simpl 473	. . . . . 6 |- ( ( s = S /\ e = A ) -> s = S )
9		simpr 477	. . . . . . . . 9 |- ( ( s = S /\ e = A ) -> e = A )
10	9	sneqd 4189	. . . . . . . 8 |- ( ( s = S /\ e = A ) -> { e } = { A } )
11	10	dmeqd 5326	. . . . . . 7 |- ( ( s = S /\ e = A ) -> dom { e } = dom { A } )
12	11	difeq2d 3728	. . . . . 6 |- ( ( s = S /\ e = A ) -> ( _V \ dom { e } ) = ( _V \ dom { A } ) )
13	8, 12	reseq12d 5397	. . . . 5 |- ( ( s = S /\ e = A ) -> ( s |` ( _V \ dom { e } ) ) = ( S |` ( _V \ dom { A } ) ) )
14	13, 10	uneq12d 3768	. . . 4 |- ( ( s = S /\ e = A ) -> ( ( s |` ( _V \ dom { e } ) ) u. { e } ) = ( ( S |` ( _V \ dom { A } ) ) u. { A } ) )
15		df-sets 15864	. . . 4 |- sSet = ( s e. _V , e e. _V |-> ( ( s |` ( _V \ dom { e } ) ) u. { e } ) )
16	14, 15	ovmpt2ga 6790	. . 3 |- ( ( S e. _V /\ A e. _V /\ ( ( S |` ( _V \ dom { A } ) ) u. { A } ) e. _V ) -> ( S sSet A ) = ( ( S |` ( _V \ dom { A } ) ) u. { A } ) )
17	7, 16	mpd3an3 1425	. 2 |- ( ( S e. _V /\ A e. _V ) -> ( S sSet A ) = ( ( S |` ( _V \ dom { A } ) ) u. { A } ) )
18	1, 2, 17	syl2an 494	1 |- ( ( S e. V /\ A e. W ) -> ( S sSet A ) = ( ( S |` ( _V \ dom { A } ) ) u. { A } ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   u. cun 3572   {csn 4177   dom cdm 5114   |` cres 5116  (class class class)co 6650   sSet csts 15855

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-8 1992  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  ax-un 6949

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-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-sets 15864

This theorem is referenced by:  setsval  15888  setsdm  15892  setsfun  15893  setsfun0  15894  wunsets  15900  setsres  15901