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Theorem prf1 15671
Description: Value of the pairing functor on objects. (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 )
Assertion
Ref Expression
prf1  |-  ( ph  ->  ( ( 1st `  P
) `  X )  =  <. ( ( 1st `  F ) `  X
) ,  ( ( 1st `  G ) `
 X ) >.
)

Proof of Theorem prf1
Dummy variables  h  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 15670 . . 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 5858 . . . . . 6  |-  ( Base `  C )  e.  _V
82, 7eqeltri 2538 . . . . 5  |-  B  e. 
_V
98mptex 6118 . . . 4  |-  ( x  e.  B  |->  <. (
( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. )  e.  _V
108, 8mpt2ex 6850 . . . 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, 10op1std 6783 . . 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
) >. ) ) >.  ->  ( 1st `  P
)  =  ( x  e.  B  |->  <. (
( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. ) )
126, 11syl 16 . 2  |-  ( ph  ->  ( 1st `  P
)  =  ( x  e.  B  |->  <. (
( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. ) )
13 simpr 459 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  x  =  X )
1413fveq2d 5852 . . 3  |-  ( (
ph  /\  x  =  X )  ->  (
( 1st `  F
) `  x )  =  ( ( 1st `  F ) `  X
) )
1513fveq2d 5852 . . 3  |-  ( (
ph  /\  x  =  X )  ->  (
( 1st `  G
) `  x )  =  ( ( 1st `  G ) `  X
) )
1614, 15opeq12d 4211 . 2  |-  ( (
ph  /\  x  =  X )  ->  <. (
( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >.  =  <. (
( 1st `  F
) `  X ) ,  ( ( 1st `  G ) `  X
) >. )
17 prf1.x . 2  |-  ( ph  ->  X  e.  B )
18 opex 4701 . . 3  |-  <. (
( 1st `  F
) `  X ) ,  ( ( 1st `  G ) `  X
) >.  e.  _V
1918a1i 11 . 2  |-  ( ph  -> 
<. ( ( 1st `  F
) `  X ) ,  ( ( 1st `  G ) `  X
) >.  e.  _V )
2012, 16, 17, 19fvmptd 5936 1  |-  ( ph  ->  ( ( 1st `  P
) `  X )  =  <. ( ( 1st `  F ) `  X
) ,  ( ( 1st `  G ) `
 X ) >.
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   _Vcvv 3106   <.cop 4022    |-> cmpt 4497   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   1stc1st 6771   2ndc2nd 6772   Basecbs 14719   Hom chom 14798    Func cfunc 15345   ⟨,⟩F cprf 15642
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 15349  df-prf 15646
This theorem is referenced by:  prfcl  15674  uncf1  15707  uncf2  15708  yonedalem21  15744  yonedalem22  15749
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