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Theorem dpjval 16574
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  =  ( proj1 `  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  =  ( proj1 `  G )
51, 2, 3, 4dpjfval 16573 . 2  |-  ( ph  ->  P  =  ( x  e.  I  |->  ( ( S `  x ) Q ( G DProd  ( S  |`  ( I  \  { x } ) ) ) ) ) )
6 simpr 461 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  x  =  X )
76fveq2d 5714 . . 3  |-  ( (
ph  /\  x  =  X )  ->  ( S `  x )  =  ( S `  X ) )
86sneqd 3908 . . . . . 6  |-  ( (
ph  /\  x  =  X )  ->  { x }  =  { X } )
98difeq2d 3493 . . . . 5  |-  ( (
ph  /\  x  =  X )  ->  (
I  \  { x } )  =  ( I  \  { X } ) )
109reseq2d 5129 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  ( S  |`  ( I  \  { x } ) )  =  ( S  |`  ( I  \  { X } ) ) )
1110oveq2d 6126 . . 3  |-  ( (
ph  /\  x  =  X )  ->  ( G DProd  ( S  |`  (
I  \  { x } ) ) )  =  ( G DProd  ( S  |`  ( I  \  { X } ) ) ) )
127, 11oveq12d 6128 . 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 6135 . . 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 5798 1  |-  ( ph  ->  ( P `  X
)  =  ( ( S `  X ) Q ( G DProd  ( S  |`  ( I  \  { X } ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2991    \ cdif 3344   {csn 3896   class class class wbr 4311   dom cdm 4859    |` cres 4861   ` cfv 5437  (class class class)co 6110   proj1cpj1 16153   DProd cdprd 16494  dProjcdpj 16495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4422  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-reu 2741  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-id 4655  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6596  df-2nd 6597  df-ixp 7283  df-dprd 16496  df-dpj 16497
This theorem is referenced by:  dpjf  16575  dpjidcl  16576  dpjlid  16579  dpjghm  16581  dpjidclOLD  16583
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