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Theorem prf2 15670
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 15669 . 2  |-  ( ph  ->  ( X ( 2nd `  P ) Y )  =  ( h  e.  ( X H Y )  |->  <. ( ( X ( 2nd `  F
) Y ) `  h ) ,  ( ( X ( 2nd `  G ) Y ) `
 h ) >.
) )
9 simpr 459 . . . 4  |-  ( (
ph  /\  h  =  K )  ->  h  =  K )
109fveq2d 5852 . . 3  |-  ( (
ph  /\  h  =  K )  ->  (
( X ( 2nd `  F ) Y ) `
 h )  =  ( ( X ( 2nd `  F ) Y ) `  K
) )
119fveq2d 5852 . . 3  |-  ( (
ph  /\  h  =  K )  ->  (
( X ( 2nd `  G ) Y ) `
 h )  =  ( ( X ( 2nd `  G ) Y ) `  K
) )
1210, 11opeq12d 4211 . 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 4701 . . 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 5936 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 367    = wceq 1398    e. wcel 1823   _Vcvv 3106   <.cop 4022   ` cfv 5570  (class class class)co 6270   2ndc2nd 6772   Basecbs 14716   Hom chom 14795    Func cfunc 15342   ⟨,⟩F cprf 15639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-map 7414  df-ixp 7463  df-func 15346  df-prf 15643
This theorem is referenced by:  prfcl  15671  prf1st  15672  prf2nd  15673  uncf2  15705  yonedalem22  15746
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