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Theorem dpjval 16545
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 16544 . 2  |-  ( ph  ->  P  =  ( x  e.  I  |->  ( ( S `  x ) Q ( G DProd  ( S  |`  ( I  \  { x } ) ) ) ) ) )
6 simpr 458 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  x  =  X )
76fveq2d 5692 . . 3  |-  ( (
ph  /\  x  =  X )  ->  ( S `  x )  =  ( S `  X ) )
86sneqd 3886 . . . . . 6  |-  ( (
ph  /\  x  =  X )  ->  { x }  =  { X } )
98difeq2d 3471 . . . . 5  |-  ( (
ph  /\  x  =  X )  ->  (
I  \  { x } )  =  ( I  \  { X } ) )
109reseq2d 5106 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  ( S  |`  ( I  \  { x } ) )  =  ( S  |`  ( I  \  { X } ) ) )
1110oveq2d 6106 . . 3  |-  ( (
ph  /\  x  =  X )  ->  ( G DProd  ( S  |`  (
I  \  { x } ) ) )  =  ( G DProd  ( S  |`  ( I  \  { X } ) ) ) )
127, 11oveq12d 6108 . 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 6115 . . 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 5776 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 1364    e. wcel 1761   _Vcvv 2970    \ cdif 3322   {csn 3874   class class class wbr 4289   dom cdm 4836    |` cres 4838   ` cfv 5415  (class class class)co 6090   proj1cpj1 16127   DProd cdprd 16465  dProjcdpj 16466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-ixp 7260  df-dprd 16467  df-dpj 16468
This theorem is referenced by:  dpjf  16546  dpjidcl  16547  dpjlid  16550  dpjghm  16552  dpjidclOLD  16554
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