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Theorem dpjval 15307
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 15306 . 2  |-  ( ph  ->  P  =  ( x  e.  I  |->  ( ( S `  x ) Q ( G DProd  ( S  |`  ( I  \  { x } ) ) ) ) ) )
6 simpr 447 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  x  =  X )
76fveq2d 5545 . . 3  |-  ( (
ph  /\  x  =  X )  ->  ( S `  x )  =  ( S `  X ) )
86sneqd 3666 . . . . . 6  |-  ( (
ph  /\  x  =  X )  ->  { x }  =  { X } )
98difeq2d 3307 . . . . 5  |-  ( (
ph  /\  x  =  X )  ->  (
I  \  { x } )  =  ( I  \  { X } ) )
109reseq2d 4971 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  ( S  |`  ( I  \  { x } ) )  =  ( S  |`  ( I  \  { X } ) ) )
1110oveq2d 5890 . . 3  |-  ( (
ph  /\  x  =  X )  ->  ( G DProd  ( S  |`  (
I  \  { x } ) ) )  =  ( G DProd  ( S  |`  ( I  \  { X } ) ) ) )
127, 11oveq12d 5892 . 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 5899 . . 3  |-  ( ( S `  X ) Q ( G DProd  ( S  |`  ( I  \  { X } ) ) ) )  e.  _V
1514a1i 10 . 2  |-  ( ph  ->  ( ( S `  X ) Q ( G DProd  ( S  |`  ( I  \  { X } ) ) ) )  e.  _V )
165, 12, 13, 15fvmptd 5622 1  |-  ( ph  ->  ( P `  X
)  =  ( ( S `  X ) Q ( G DProd  ( S  |`  ( I  \  { X } ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    \ cdif 3162   {csn 3653   class class class wbr 4039   dom cdm 4705    |` cres 4707   ` cfv 5271  (class class class)co 5874   proj 1cpj1 14962   DProd cdprd 15247  dProjcdpj 15248
This theorem is referenced by:  dpjf  15308  dpjidcl  15309  dpjlid  15312  dpjghm  15314
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-ixp 6834  df-dprd 15249  df-dpj 15250
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