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Theorem setsvalg 13171
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 2796 . 2  |-  ( S  e.  V  ->  S  e.  _V )
2 elex 2796 . 2  |-  ( A  e.  W  ->  A  e.  _V )
3 resexg 4994 . . . . 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 4216 . . . 4  |-  { A }  e.  _V
6 unexg 4521 . . . 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 3653 . . . . . . . 8  |-  ( ( s  =  S  /\  e  =  A )  ->  { e }  =  { A } )
1110dmeqd 4881 . . . . . . 7  |-  ( ( s  =  S  /\  e  =  A )  ->  dom  { e }  =  dom  { A } )
1211difeq2d 3294 . . . . . 6  |-  ( ( s  =  S  /\  e  =  A )  ->  ( _V  \  dom  { e } )  =  ( _V  \  dom  { A } ) )
138, 12reseq12d 4956 . . . . 5  |-  ( ( s  =  S  /\  e  =  A )  ->  ( s  |`  ( _V  \  dom  { e } ) )  =  ( S  |`  ( _V  \  dom  { A } ) ) )
1413, 10uneq12d 3330 . . . 4  |-  ( ( s  =  S  /\  e  =  A )  ->  ( ( s  |`  ( _V  \  dom  {
e } ) )  u.  { e } )  =  ( ( S  |`  ( _V  \  dom  { A }
) )  u.  { A } ) )
15 df-sets 13154 . . . 4  |- sSet  =  ( s  e.  _V , 
e  e.  _V  |->  ( ( s  |`  ( _V  \  dom  { e } ) )  u. 
{ e } ) )
1614, 15ovmpt2ga 5977 . . 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 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149    u. cun 3150   {csn 3640   dom cdm 4689    |` cres 4691  (class class class)co 5858   sSet csts 13146
This theorem is referenced by:  setsval  13172  wunsets  13173  setsres  13174
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-res 4701  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-sets 13154
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