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Theorem prf2fval 13975
Description: Value of the pairing functor on morphisms. (Contributed by Mario Carneiro, 12-Jan-2017.)
Hypotheses
Ref Expression
prfval.k  |-  P  =  ( F ⟨,⟩F  G )
prfval.b  |-  B  =  ( Base `  C
)
prfval.h  |-  H  =  (  Hom  `  C
)
prfval.c  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
prfval.d  |-  ( ph  ->  G  e.  ( C 
Func  E ) )
prf1.x  |-  ( ph  ->  X  e.  B )
prf2.y  |-  ( ph  ->  Y  e.  B )
Assertion
Ref Expression
prf2fval  |-  ( ph  ->  ( X ( 2nd `  P ) Y )  =  ( h  e.  ( X H Y )  |->  <. ( ( X ( 2nd `  F
) Y ) `  h ) ,  ( ( X ( 2nd `  G ) Y ) `
 h ) >.
) )
Distinct variable groups:    B, h    h, F    ph, h    h, G    h, X    h, Y    h, H
Allowed substitution hints:    C( h)    D( h)    P( h)    E( h)

Proof of Theorem prf2fval
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prfval.k . . . 4  |-  P  =  ( F ⟨,⟩F  G )
2 prfval.b . . . 4  |-  B  =  ( Base `  C
)
3 prfval.h . . . 4  |-  H  =  (  Hom  `  C
)
4 prfval.c . . . 4  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
5 prfval.d . . . 4  |-  ( ph  ->  G  e.  ( C 
Func  E ) )
61, 2, 3, 4, 5prfval 13973 . . 3  |-  ( ph  ->  P  =  <. (
x  e.  B  |->  <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. ) ,  ( x  e.  B , 
y  e.  B  |->  ( h  e.  ( x H y )  |->  <.
( ( x ( 2nd `  F ) y ) `  h
) ,  ( ( x ( 2nd `  G
) y ) `  h ) >. )
) >. )
7 fvex 5539 . . . . . 6  |-  ( Base `  C )  e.  _V
82, 7eqeltri 2353 . . . . 5  |-  B  e. 
_V
98mptex 5746 . . . 4  |-  ( x  e.  B  |->  <. (
( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. )  e.  _V
108, 8mpt2ex 6198 . . . 4  |-  ( x  e.  B ,  y  e.  B  |->  ( h  e.  ( x H y )  |->  <. (
( x ( 2nd `  F ) y ) `
 h ) ,  ( ( x ( 2nd `  G ) y ) `  h
) >. ) )  e. 
_V
119, 10op2ndd 6131 . . 3  |-  ( P  =  <. ( x  e.  B  |->  <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
) ,  ( x  e.  B ,  y  e.  B  |->  ( h  e.  ( x H y )  |->  <. (
( x ( 2nd `  F ) y ) `
 h ) ,  ( ( x ( 2nd `  G ) y ) `  h
) >. ) ) >.  ->  ( 2nd `  P
)  =  ( x  e.  B ,  y  e.  B  |->  ( h  e.  ( x H y )  |->  <. (
( x ( 2nd `  F ) y ) `
 h ) ,  ( ( x ( 2nd `  G ) y ) `  h
) >. ) ) )
126, 11syl 15 . 2  |-  ( ph  ->  ( 2nd `  P
)  =  ( x  e.  B ,  y  e.  B  |->  ( h  e.  ( x H y )  |->  <. (
( x ( 2nd `  F ) y ) `
 h ) ,  ( ( x ( 2nd `  G ) y ) `  h
) >. ) ) )
13 simprl 732 . . . 4  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  x  =  X )
14 simprr 733 . . . 4  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
y  =  Y )
1513, 14oveq12d 5876 . . 3  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( x H y )  =  ( X H Y ) )
1613, 14oveq12d 5876 . . . . 5  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( x ( 2nd `  F ) y )  =  ( X ( 2nd `  F ) Y ) )
1716fveq1d 5527 . . . 4  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( ( x ( 2nd `  F ) y ) `  h
)  =  ( ( X ( 2nd `  F
) Y ) `  h ) )
1813, 14oveq12d 5876 . . . . 5  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( x ( 2nd `  G ) y )  =  ( X ( 2nd `  G ) Y ) )
1918fveq1d 5527 . . . 4  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( ( x ( 2nd `  G ) y ) `  h
)  =  ( ( X ( 2nd `  G
) Y ) `  h ) )
2017, 19opeq12d 3804 . . 3  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  <. ( ( x ( 2nd `  F ) y ) `  h
) ,  ( ( x ( 2nd `  G
) y ) `  h ) >.  =  <. ( ( X ( 2nd `  F ) Y ) `
 h ) ,  ( ( X ( 2nd `  G ) Y ) `  h
) >. )
2115, 20mpteq12dv 4098 . 2  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( h  e.  ( x H y ) 
|->  <. ( ( x ( 2nd `  F
) y ) `  h ) ,  ( ( x ( 2nd `  G ) y ) `
 h ) >.
)  =  ( h  e.  ( X H Y )  |->  <. (
( X ( 2nd `  F ) Y ) `
 h ) ,  ( ( X ( 2nd `  G ) Y ) `  h
) >. ) )
22 prf1.x . 2  |-  ( ph  ->  X  e.  B )
23 prf2.y . 2  |-  ( ph  ->  Y  e.  B )
24 ovex 5883 . . . 4  |-  ( X H Y )  e. 
_V
2524mptex 5746 . . 3  |-  ( h  e.  ( X H Y )  |->  <. (
( X ( 2nd `  F ) Y ) `
 h ) ,  ( ( X ( 2nd `  G ) Y ) `  h
) >. )  e.  _V
2625a1i 10 . 2  |-  ( ph  ->  ( h  e.  ( X H Y ) 
|->  <. ( ( X ( 2nd `  F
) Y ) `  h ) ,  ( ( X ( 2nd `  G ) Y ) `
 h ) >.
)  e.  _V )
2712, 21, 22, 23, 26ovmpt2d 5975 1  |-  ( ph  ->  ( X ( 2nd `  P ) Y )  =  ( h  e.  ( X H Y )  |->  <. ( ( X ( 2nd `  F
) Y ) `  h ) ,  ( ( X ( 2nd `  G ) Y ) `
 h ) >.
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643    e. cmpt 4077   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1stc1st 6120   2ndc2nd 6121   Basecbs 13148    Hom chom 13219    Func cfunc 13728   ⟨,⟩F cprf 13945
This theorem is referenced by:  prf2  13976
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774  df-ixp 6818  df-func 13732  df-prf 13949
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