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Theorem setsvalg 12545
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 2763 . 2  |-  ( S  e.  V  ->  S  e.  _V )
2 elex 2763 . 2  |-  ( A  e.  W  ->  A  e.  _V )
3 resexg 4965 . . . 4  |-  ( S  e.  _V  ->  ( S  |`  ( _V  \  dom  { A } ) )  e.  _V )
4 snexg 4202 . . . 4  |-  ( A  e.  _V  ->  { A }  e.  _V )
5 unexg 4461 . . . 4  |-  ( ( ( S  |`  ( _V  \  dom  { A } ) )  e. 
_V  /\  { A }  e.  _V )  ->  ( ( S  |`  ( _V  \  dom  { A } ) )  u. 
{ A } )  e.  _V )
63, 4, 5syl2an 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 )
98sneqd 3620 . . . . . . . 8  |-  ( ( s  =  S  /\  e  =  A )  ->  { e }  =  { A } )
109dmeqd 4847 . . . . . . 7  |-  ( ( s  =  S  /\  e  =  A )  ->  dom  { e }  =  dom  { A } )
1110difeq2d 3268 . . . . . 6  |-  ( ( s  =  S  /\  e  =  A )  ->  ( _V  \  dom  { e } )  =  ( _V  \  dom  { A } ) )
127, 11reseq12d 4926 . . . . 5  |-  ( ( s  =  S  /\  e  =  A )  ->  ( s  |`  ( _V  \  dom  { e } ) )  =  ( S  |`  ( _V  \  dom  { A } ) ) )
1312, 9uneq12d 3305 . . . 4  |-  ( ( s  =  S  /\  e  =  A )  ->  ( ( s  |`  ( _V  \  dom  {
e } ) )  u.  { e } )  =  ( ( S  |`  ( _V  \  dom  { A }
) )  u.  { A } ) )
14 df-sets 12522 . . . 4  |- sSet  =  ( s  e.  _V , 
e  e.  _V  |->  ( ( s  |`  ( _V  \  dom  { e } ) )  u. 
{ e } ) )
1513, 14ovmpoga 6027 . . 3  |-  ( ( S  e.  _V  /\  A  e.  _V  /\  (
( S  |`  ( _V  \  dom  { A } ) )  u. 
{ A } )  e.  _V )  -> 
( S sSet  A )  =  ( ( S  |`  ( _V  \  dom  { A } ) )  u.  { A }
) )
166, 15mpd3an3 1349 . 2  |-  ( ( S  e.  _V  /\  A  e.  _V )  ->  ( S sSet  A )  =  ( ( S  |`  ( _V  \  dom  { A } ) )  u.  { A }
) )
171, 2, 16syl2an 289 1  |-  ( ( S  e.  V  /\  A  e.  W )  ->  ( S sSet  A )  =  ( ( S  |`  ( _V  \  dom  { A } ) )  u.  { A }
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160   _Vcvv 2752    \ cdif 3141    u. cun 3142   {csn 3607   dom cdm 4644    |` cres 4646  (class class class)co 5897   sSet csts 12513
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-res 4656  df-iota 5196  df-fun 5237  df-fv 5243  df-ov 5900  df-oprab 5901  df-mpo 5902  df-sets 12522
This theorem is referenced by:  setsvala  12546  setsfun  12550  setsfun0  12551  setsresg  12553
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