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Theorem prf1 14260
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 14259 . . 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 5709 . . . . . 6  |-  ( Base `  C )  e.  _V
82, 7eqeltri 2482 . . . . 5  |-  B  e. 
_V
98mptex 5933 . . . 4  |-  ( x  e.  B  |->  <. (
( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. )  e.  _V
108, 8mpt2ex 6392 . . . 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 6324 . . 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 448 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  x  =  X )
1413fveq2d 5699 . . 3  |-  ( (
ph  /\  x  =  X )  ->  (
( 1st `  F
) `  x )  =  ( ( 1st `  F ) `  X
) )
1513fveq2d 5699 . . 3  |-  ( (
ph  /\  x  =  X )  ->  (
( 1st `  G
) `  x )  =  ( ( 1st `  G ) `  X
) )
1614, 15opeq12d 3960 . 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 4395 . . 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 5777 1  |-  ( ph  ->  ( ( 1st `  P
) `  X )  =  <. ( ( 1st `  F ) `  X
) ,  ( ( 1st `  G ) `
 X ) >.
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2924   <.cop 3785    e. cmpt 4234   ` cfv 5421  (class class class)co 6048    e. cmpt2 6050   1stc1st 6314   2ndc2nd 6315   Basecbs 13432    Hom chom 13503    Func cfunc 14014   ⟨,⟩F cprf 14231
This theorem is referenced by:  prfcl  14263  uncf1  14296  uncf2  14297  yonedalem21  14333  yonedalem22  14338
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-map 6987  df-ixp 7031  df-func 14018  df-prf 14235
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