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Theorem prf1 14328
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 14327 . . 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 5771 . . . . . 6  |-  ( Base `  C )  e.  _V
82, 7eqeltri 2512 . . . . 5  |-  B  e. 
_V
98mptex 5995 . . . 4  |-  ( x  e.  B  |->  <. (
( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. )  e.  _V
108, 8mpt2ex 6454 . . . 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 6386 . . 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 449 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  x  =  X )
1413fveq2d 5761 . . 3  |-  ( (
ph  /\  x  =  X )  ->  (
( 1st `  F
) `  x )  =  ( ( 1st `  F ) `  X
) )
1513fveq2d 5761 . . 3  |-  ( (
ph  /\  x  =  X )  ->  (
( 1st `  G
) `  x )  =  ( ( 1st `  G ) `  X
) )
1614, 15opeq12d 4016 . 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 4456 . . 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 5839 1  |-  ( ph  ->  ( ( 1st `  P
) `  X )  =  <. ( ( 1st `  F ) `  X
) ,  ( ( 1st `  G ) `
 X ) >.
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1727   _Vcvv 2962   <.cop 3841    e. cmpt 4291   ` cfv 5483  (class class class)co 6110    e. cmpt2 6112   1stc1st 6376   2ndc2nd 6377   Basecbs 13500    Hom chom 13571    Func cfunc 14082   ⟨,⟩F cprf 14299
This theorem is referenced by:  prfcl  14331  uncf1  14364  uncf2  14365  yonedalem21  14401  yonedalem22  14406
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-map 7049  df-ixp 7093  df-func 14086  df-prf 14303
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