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Theorem prf2 14291
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 14290 . 2  |-  ( ph  ->  ( X ( 2nd `  P ) Y )  =  ( h  e.  ( X H Y )  |->  <. ( ( X ( 2nd `  F
) Y ) `  h ) ,  ( ( X ( 2nd `  G ) Y ) `
 h ) >.
) )
9 simpr 448 . . . 4  |-  ( (
ph  /\  h  =  K )  ->  h  =  K )
109fveq2d 5724 . . 3  |-  ( (
ph  /\  h  =  K )  ->  (
( X ( 2nd `  F ) Y ) `
 h )  =  ( ( X ( 2nd `  F ) Y ) `  K
) )
119fveq2d 5724 . . 3  |-  ( (
ph  /\  h  =  K )  ->  (
( X ( 2nd `  G ) Y ) `
 h )  =  ( ( X ( 2nd `  G ) Y ) `  K
) )
1210, 11opeq12d 3984 . 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 4419 . . 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 5802 1  |-  ( ph  ->  ( ( X ( 2nd `  P ) Y ) `  K
)  =  <. (
( X ( 2nd `  F ) Y ) `
 K ) ,  ( ( X ( 2nd `  G ) Y ) `  K
) >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948   <.cop 3809   ` cfv 5446  (class class class)co 6073   2ndc2nd 6340   Basecbs 13461    Hom chom 13532    Func cfunc 14043   ⟨,⟩F cprf 14260
This theorem is referenced by:  prfcl  14292  prf1st  14293  prf2nd  14294  uncf2  14326  yonedalem22  14367
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-map 7012  df-ixp 7056  df-func 14047  df-prf 14264
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