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Theorem setsvalg 14502
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.
StepHypRef Expression
1 elex 3115 . 2  |-  ( S  e.  V  ->  S  e.  _V )
2 elex 3115 . 2  |-  ( A  e.  W  ->  A  e.  _V )
3 resexg 5307 . . . . 5  |-  ( S  e.  _V  ->  ( S  |`  ( _V  \  dom  { A } ) )  e.  _V )
43adantr 465 . . . 4  |-  ( ( S  e.  _V  /\  A  e.  _V )  ->  ( S  |`  ( _V  \  dom  { A } ) )  e. 
_V )
5 snex 4681 . . . 4  |-  { A }  e.  _V
6 unexg 6576 . . . 4  |-  ( ( ( S  |`  ( _V  \  dom  { A } ) )  e. 
_V  /\  { A }  e.  _V )  ->  ( ( S  |`  ( _V  \  dom  { A } ) )  u. 
{ A } )  e.  _V )
74, 5, 6sylancl 662 . . 3  |-  ( ( S  e.  _V  /\  A  e.  _V )  ->  ( ( S  |`  ( _V  \  dom  { A } ) )  u. 
{ A } )  e.  _V )
8 simpl 457 . . . . . 6  |-  ( ( s  =  S  /\  e  =  A )  ->  s  =  S )
9 simpr 461 . . . . . . . . 9  |-  ( ( s  =  S  /\  e  =  A )  ->  e  =  A )
109sneqd 4032 . . . . . . . 8  |-  ( ( s  =  S  /\  e  =  A )  ->  { e }  =  { A } )
1110dmeqd 5196 . . . . . . 7  |-  ( ( s  =  S  /\  e  =  A )  ->  dom  { e }  =  dom  { A } )
1211difeq2d 3615 . . . . . 6  |-  ( ( s  =  S  /\  e  =  A )  ->  ( _V  \  dom  { e } )  =  ( _V  \  dom  { A } ) )
138, 12reseq12d 5265 . . . . 5  |-  ( ( s  =  S  /\  e  =  A )  ->  ( s  |`  ( _V  \  dom  { e } ) )  =  ( S  |`  ( _V  \  dom  { A } ) ) )
1413, 10uneq12d 3652 . . . 4  |-  ( ( s  =  S  /\  e  =  A )  ->  ( ( s  |`  ( _V  \  dom  {
e } ) )  u.  { e } )  =  ( ( S  |`  ( _V  \  dom  { A }
) )  u.  { A } ) )
15 df-sets 14485 . . . 4  |- sSet  =  ( s  e.  _V , 
e  e.  _V  |->  ( ( s  |`  ( _V  \  dom  { e } ) )  u. 
{ e } ) )
1614, 15ovmpt2ga 6407 . . 3  |-  ( ( S  e.  _V  /\  A  e.  _V  /\  (
( S  |`  ( _V  \  dom  { A } ) )  u. 
{ A } )  e.  _V )  -> 
( S sSet  A )  =  ( ( S  |`  ( _V  \  dom  { A } ) )  u.  { A }
) )
177, 16mpd3an3 1320 . 2  |-  ( ( S  e.  _V  /\  A  e.  _V )  ->  ( S sSet  A )  =  ( ( S  |`  ( _V  \  dom  { A } ) )  u.  { A }
) )
181, 2, 17syl2an 477 1  |-  ( ( S  e.  V  /\  A  e.  W )  ->  ( S sSet  A )  =  ( ( S  |`  ( _V  \  dom  { A } ) )  u.  { A }
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   _Vcvv 3106    \ cdif 3466    u. cun 3467   {csn 4020   dom cdm 4992    |` cres 4994  (class class class)co 6275   sSet csts 14477
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-res 5004  df-iota 5542  df-fun 5581  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-sets 14485
This theorem is referenced by:  setsval  14503  wunsets  14506  setsres  14507
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