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Theorem prf2 15123
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 )
prf2.k  |-  ( ph  ->  K  e.  ( X H Y ) )
Assertion
Ref Expression
prf2  |-  ( ph  ->  ( ( X ( 2nd `  P ) Y ) `  K
)  =  <. (
( X ( 2nd `  F ) Y ) `
 K ) ,  ( ( X ( 2nd `  G ) Y ) `  K
) >. )

Proof of Theorem prf2
Dummy variable  h is distinct from all other variables.
StepHypRef Expression
1 prfval.k . . 3  |-  P  =  ( F ⟨,⟩F  G )
2 prfval.b . . 3  |-  B  =  ( Base `  C
)
3 prfval.h . . 3  |-  H  =  ( Hom  `  C
)
4 prfval.c . . 3  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
5 prfval.d . . 3  |-  ( ph  ->  G  e.  ( C 
Func  E ) )
6 prf1.x . . 3  |-  ( ph  ->  X  e.  B )
7 prf2.y . . 3  |-  ( ph  ->  Y  e.  B )
81, 2, 3, 4, 5, 6, 7prf2fval 15122 . 2  |-  ( ph  ->  ( X ( 2nd `  P ) Y )  =  ( h  e.  ( X H Y )  |->  <. ( ( X ( 2nd `  F
) Y ) `  h ) ,  ( ( X ( 2nd `  G ) Y ) `
 h ) >.
) )
9 simpr 461 . . . 4  |-  ( (
ph  /\  h  =  K )  ->  h  =  K )
109fveq2d 5796 . . 3  |-  ( (
ph  /\  h  =  K )  ->  (
( X ( 2nd `  F ) Y ) `
 h )  =  ( ( X ( 2nd `  F ) Y ) `  K
) )
119fveq2d 5796 . . 3  |-  ( (
ph  /\  h  =  K )  ->  (
( X ( 2nd `  G ) Y ) `
 h )  =  ( ( X ( 2nd `  G ) Y ) `  K
) )
1210, 11opeq12d 4168 . 2  |-  ( (
ph  /\  h  =  K )  ->  <. (
( X ( 2nd `  F ) Y ) `
 h ) ,  ( ( X ( 2nd `  G ) Y ) `  h
) >.  =  <. (
( X ( 2nd `  F ) Y ) `
 K ) ,  ( ( X ( 2nd `  G ) Y ) `  K
) >. )
13 prf2.k . 2  |-  ( ph  ->  K  e.  ( X H Y ) )
14 opex 4657 . . 3  |-  <. (
( X ( 2nd `  F ) Y ) `
 K ) ,  ( ( X ( 2nd `  G ) Y ) `  K
) >.  e.  _V
1514a1i 11 . 2  |-  ( ph  -> 
<. ( ( X ( 2nd `  F ) Y ) `  K
) ,  ( ( X ( 2nd `  G
) Y ) `  K ) >.  e.  _V )
168, 12, 13, 15fvmptd 5881 1  |-  ( ph  ->  ( ( X ( 2nd `  P ) Y ) `  K
)  =  <. (
( X ( 2nd `  F ) Y ) `
 K ) ,  ( ( X ( 2nd `  G ) Y ) `  K
) >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3071   <.cop 3984   ` cfv 5519  (class class class)co 6193   2ndc2nd 6679   Basecbs 14285   Hom chom 14360    Func cfunc 14875   ⟨,⟩F cprf 15092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-1st 6680  df-2nd 6681  df-map 7319  df-ixp 7367  df-func 14879  df-prf 15096
This theorem is referenced by:  prfcl  15124  prf1st  15125  prf2nd  15126  uncf2  15158  yonedalem22  15199
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