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Theorem setsvalg 13187
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 2809 . 2  |-  ( S  e.  V  ->  S  e.  _V )
2 elex 2809 . 2  |-  ( A  e.  W  ->  A  e.  _V )
3 resexg 5010 . . . . 5  |-  ( S  e.  _V  ->  ( S  |`  ( _V  \  dom  { A } ) )  e.  _V )
43adantr 451 . . . 4  |-  ( ( S  e.  _V  /\  A  e.  _V )  ->  ( S  |`  ( _V  \  dom  { A } ) )  e. 
_V )
5 snex 4232 . . . 4  |-  { A }  e.  _V
6 unexg 4537 . . . 4  |-  ( ( ( S  |`  ( _V  \  dom  { A } ) )  e. 
_V  /\  { A }  e.  _V )  ->  ( ( S  |`  ( _V  \  dom  { A } ) )  u. 
{ A } )  e.  _V )
74, 5, 6sylancl 643 . . 3  |-  ( ( S  e.  _V  /\  A  e.  _V )  ->  ( ( S  |`  ( _V  \  dom  { A } ) )  u. 
{ A } )  e.  _V )
8 simpl 443 . . . . . 6  |-  ( ( s  =  S  /\  e  =  A )  ->  s  =  S )
9 simpr 447 . . . . . . . . 9  |-  ( ( s  =  S  /\  e  =  A )  ->  e  =  A )
109sneqd 3666 . . . . . . . 8  |-  ( ( s  =  S  /\  e  =  A )  ->  { e }  =  { A } )
1110dmeqd 4897 . . . . . . 7  |-  ( ( s  =  S  /\  e  =  A )  ->  dom  { e }  =  dom  { A } )
1211difeq2d 3307 . . . . . 6  |-  ( ( s  =  S  /\  e  =  A )  ->  ( _V  \  dom  { e } )  =  ( _V  \  dom  { A } ) )
138, 12reseq12d 4972 . . . . 5  |-  ( ( s  =  S  /\  e  =  A )  ->  ( s  |`  ( _V  \  dom  { e } ) )  =  ( S  |`  ( _V  \  dom  { A } ) ) )
1413, 10uneq12d 3343 . . . 4  |-  ( ( s  =  S  /\  e  =  A )  ->  ( ( s  |`  ( _V  \  dom  {
e } ) )  u.  { e } )  =  ( ( S  |`  ( _V  \  dom  { A }
) )  u.  { A } ) )
15 df-sets 13170 . . . 4  |- sSet  =  ( s  e.  _V , 
e  e.  _V  |->  ( ( s  |`  ( _V  \  dom  { e } ) )  u. 
{ e } ) )
1614, 15ovmpt2ga 5993 . . 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 1278 . 2  |-  ( ( S  e.  _V  /\  A  e.  _V )  ->  ( S sSet  A )  =  ( ( S  |`  ( _V  \  dom  { A } ) )  u.  { A }
) )
181, 2, 17syl2an 463 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 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    \ cdif 3162    u. cun 3163   {csn 3653   dom cdm 4705    |` cres 4707  (class class class)co 5874   sSet csts 13162
This theorem is referenced by:  setsval  13188  wunsets  13189  setsres  13190
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-res 4717  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-sets 13170
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