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Theorem dpjval 15616
Description: Value of the direct product projection (defined in terms of binary projection). (Contributed by Mario Carneiro, 26-Apr-2016.)
Hypotheses
Ref Expression
dpjfval.1  |-  ( ph  ->  G dom DProd  S )
dpjfval.2  |-  ( ph  ->  dom  S  =  I )
dpjfval.p  |-  P  =  ( GdProj S )
dpjfval.q  |-  Q  =  ( proj 1 `  G )
dpjval.3  |-  ( ph  ->  X  e.  I )
Assertion
Ref Expression
dpjval  |-  ( ph  ->  ( P `  X
)  =  ( ( S `  X ) Q ( G DProd  ( S  |`  ( I  \  { X } ) ) ) ) )

Proof of Theorem dpjval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dpjfval.1 . . 3  |-  ( ph  ->  G dom DProd  S )
2 dpjfval.2 . . 3  |-  ( ph  ->  dom  S  =  I )
3 dpjfval.p . . 3  |-  P  =  ( GdProj S )
4 dpjfval.q . . 3  |-  Q  =  ( proj 1 `  G )
51, 2, 3, 4dpjfval 15615 . 2  |-  ( ph  ->  P  =  ( x  e.  I  |->  ( ( S `  x ) Q ( G DProd  ( S  |`  ( I  \  { x } ) ) ) ) ) )
6 simpr 449 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  x  =  X )
76fveq2d 5734 . . 3  |-  ( (
ph  /\  x  =  X )  ->  ( S `  x )  =  ( S `  X ) )
86sneqd 3829 . . . . . 6  |-  ( (
ph  /\  x  =  X )  ->  { x }  =  { X } )
98difeq2d 3467 . . . . 5  |-  ( (
ph  /\  x  =  X )  ->  (
I  \  { x } )  =  ( I  \  { X } ) )
109reseq2d 5148 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  ( S  |`  ( I  \  { x } ) )  =  ( S  |`  ( I  \  { X } ) ) )
1110oveq2d 6099 . . 3  |-  ( (
ph  /\  x  =  X )  ->  ( G DProd  ( S  |`  (
I  \  { x } ) ) )  =  ( G DProd  ( S  |`  ( I  \  { X } ) ) ) )
127, 11oveq12d 6101 . 2  |-  ( (
ph  /\  x  =  X )  ->  (
( S `  x
) Q ( G DProd 
( S  |`  (
I  \  { x } ) ) ) )  =  ( ( S `  X ) Q ( G DProd  ( S  |`  ( I  \  { X } ) ) ) ) )
13 dpjval.3 . 2  |-  ( ph  ->  X  e.  I )
14 ovex 6108 . . 3  |-  ( ( S `  X ) Q ( G DProd  ( S  |`  ( I  \  { X } ) ) ) )  e.  _V
1514a1i 11 . 2  |-  ( ph  ->  ( ( S `  X ) Q ( G DProd  ( S  |`  ( I  \  { X } ) ) ) )  e.  _V )
165, 12, 13, 15fvmptd 5812 1  |-  ( ph  ->  ( P `  X
)  =  ( ( S `  X ) Q ( G DProd  ( S  |`  ( I  \  { X } ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958    \ cdif 3319   {csn 3816   class class class wbr 4214   dom cdm 4880    |` cres 4882   ` cfv 5456  (class class class)co 6083   proj 1cpj1 15271   DProd cdprd 15556  dProjcdpj 15557
This theorem is referenced by:  dpjf  15617  dpjidcl  15618  dpjlid  15621  dpjghm  15623
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-ixp 7066  df-dprd 15558  df-dpj 15559
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