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Theorem dpjval 17624
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 17623 . 2  |-  ( ph  ->  P  =  ( x  e.  I  |->  ( ( S `  x ) Q ( G DProd  ( S  |`  ( I  \  { x } ) ) ) ) ) )
6 simpr 462 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  x  =  X )
76fveq2d 5885 . . 3  |-  ( (
ph  /\  x  =  X )  ->  ( S `  x )  =  ( S `  X ) )
86sneqd 4014 . . . . . 6  |-  ( (
ph  /\  x  =  X )  ->  { x }  =  { X } )
98difeq2d 3589 . . . . 5  |-  ( (
ph  /\  x  =  X )  ->  (
I  \  { x } )  =  ( I  \  { X } ) )
109reseq2d 5125 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  ( S  |`  ( I  \  { x } ) )  =  ( S  |`  ( I  \  { X } ) ) )
1110oveq2d 6321 . . 3  |-  ( (
ph  /\  x  =  X )  ->  ( G DProd  ( S  |`  (
I  \  { x } ) ) )  =  ( G DProd  ( S  |`  ( I  \  { X } ) ) ) )
127, 11oveq12d 6323 . 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 6333 . . 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 5970 1  |-  ( ph  ->  ( P `  X
)  =  ( ( S `  X ) Q ( G DProd  ( S  |`  ( I  \  { X } ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   _Vcvv 3087    \ cdif 3439   {csn 4002   class class class wbr 4426   dom cdm 4854    |` cres 4856   ` cfv 5601  (class class class)co 6305   proj1cpj1 17222   DProd cdprd 17560  dProjcdpj 17561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-ixp 7531  df-dprd 17562  df-dpj 17563
This theorem is referenced by:  dpjf  17625  dpjidcl  17626  dpjlid  17629  dpjghm  17631
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