ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  setsvalg Unicode version

Theorem setsvalg 11999
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 2697 . 2  |-  ( S  e.  V  ->  S  e.  _V )
2 elex 2697 . 2  |-  ( A  e.  W  ->  A  e.  _V )
3 resexg 4859 . . . 4  |-  ( S  e.  _V  ->  ( S  |`  ( _V  \  dom  { A } ) )  e.  _V )
4 snexg 4108 . . . 4  |-  ( A  e.  _V  ->  { A }  e.  _V )
5 unexg 4364 . . . 4  |-  ( ( ( S  |`  ( _V  \  dom  { A } ) )  e. 
_V  /\  { A }  e.  _V )  ->  ( ( S  |`  ( _V  \  dom  { A } ) )  u. 
{ A } )  e.  _V )
63, 4, 5syl2an 287 . . 3  |-  ( ( S  e.  _V  /\  A  e.  _V )  ->  ( ( S  |`  ( _V  \  dom  { A } ) )  u. 
{ A } )  e.  _V )
7 simpl 108 . . . . . 6  |-  ( ( s  =  S  /\  e  =  A )  ->  s  =  S )
8 simpr 109 . . . . . . . . 9  |-  ( ( s  =  S  /\  e  =  A )  ->  e  =  A )
98sneqd 3540 . . . . . . . 8  |-  ( ( s  =  S  /\  e  =  A )  ->  { e }  =  { A } )
109dmeqd 4741 . . . . . . 7  |-  ( ( s  =  S  /\  e  =  A )  ->  dom  { e }  =  dom  { A } )
1110difeq2d 3194 . . . . . 6  |-  ( ( s  =  S  /\  e  =  A )  ->  ( _V  \  dom  { e } )  =  ( _V  \  dom  { A } ) )
127, 11reseq12d 4820 . . . . 5  |-  ( ( s  =  S  /\  e  =  A )  ->  ( s  |`  ( _V  \  dom  { e } ) )  =  ( S  |`  ( _V  \  dom  { A } ) ) )
1312, 9uneq12d 3231 . . . 4  |-  ( ( s  =  S  /\  e  =  A )  ->  ( ( s  |`  ( _V  \  dom  {
e } ) )  u.  { e } )  =  ( ( S  |`  ( _V  \  dom  { A }
) )  u.  { A } ) )
14 df-sets 11976 . . . 4  |- sSet  =  ( s  e.  _V , 
e  e.  _V  |->  ( ( s  |`  ( _V  \  dom  { e } ) )  u. 
{ e } ) )
1513, 14ovmpoga 5900 . . 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 1316 . 2  |-  ( ( S  e.  _V  /\  A  e.  _V )  ->  ( S sSet  A )  =  ( ( S  |`  ( _V  \  dom  { A } ) )  u.  { A }
) )
171, 2, 16syl2an 287 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 103    = wceq 1331    e. wcel 1480   _Vcvv 2686    \ cdif 3068    u. cun 3069   {csn 3527   dom cdm 4539    |` cres 4541  (class class class)co 5774   sSet csts 11967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-res 4551  df-iota 5088  df-fun 5125  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-sets 11976
This theorem is referenced by:  setsvala  12000  setsfun  12004  setsfun0  12005  setsresg  12007
  Copyright terms: Public domain W3C validator