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Theorem fuccoval 13837
Description: Value of the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
fucco.q  |-  Q  =  ( C FuncCat  D )
fucco.n  |-  N  =  ( C Nat  D )
fucco.a  |-  A  =  ( Base `  C
)
fucco.o  |-  .x.  =  (comp `  D )
fucco.x  |-  .xb  =  (comp `  Q )
fucco.f  |-  ( ph  ->  R  e.  ( F N G ) )
fucco.g  |-  ( ph  ->  S  e.  ( G N H ) )
fuccoval.f  |-  ( ph  ->  X  e.  A )
Assertion
Ref Expression
fuccoval  |-  ( ph  ->  ( ( S (
<. F ,  G >.  .xb 
H ) R ) `
 X )  =  ( ( S `  X ) ( <.
( ( 1st `  F
) `  X ) ,  ( ( 1st `  G ) `  X
) >.  .x.  ( ( 1st `  H ) `  X ) ) ( R `  X ) ) )

Proof of Theorem fuccoval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fucco.q . . 3  |-  Q  =  ( C FuncCat  D )
2 fucco.n . . 3  |-  N  =  ( C Nat  D )
3 fucco.a . . 3  |-  A  =  ( Base `  C
)
4 fucco.o . . 3  |-  .x.  =  (comp `  D )
5 fucco.x . . 3  |-  .xb  =  (comp `  Q )
6 fucco.f . . 3  |-  ( ph  ->  R  e.  ( F N G ) )
7 fucco.g . . 3  |-  ( ph  ->  S  e.  ( G N H ) )
81, 2, 3, 4, 5, 6, 7fucco 13836 . 2  |-  ( ph  ->  ( S ( <. F ,  G >.  .xb 
H ) R )  =  ( x  e.  A  |->  ( ( S `
 x ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >.  .x.  ( ( 1st `  H ) `  x ) ) ( R `  x ) ) ) )
9 simpr 447 . . . . . 6  |-  ( (
ph  /\  x  =  X )  ->  x  =  X )
109fveq2d 5529 . . . . 5  |-  ( (
ph  /\  x  =  X )  ->  (
( 1st `  F
) `  x )  =  ( ( 1st `  F ) `  X
) )
119fveq2d 5529 . . . . 5  |-  ( (
ph  /\  x  =  X )  ->  (
( 1st `  G
) `  x )  =  ( ( 1st `  G ) `  X
) )
1210, 11opeq12d 3804 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  <. (
( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >.  =  <. (
( 1st `  F
) `  X ) ,  ( ( 1st `  G ) `  X
) >. )
139fveq2d 5529 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  (
( 1st `  H
) `  x )  =  ( ( 1st `  H ) `  X
) )
1412, 13oveq12d 5876 . . 3  |-  ( (
ph  /\  x  =  X )  ->  ( <. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >.  .x.  ( ( 1st `  H ) `  x ) )  =  ( <. ( ( 1st `  F ) `  X
) ,  ( ( 1st `  G ) `
 X ) >.  .x.  ( ( 1st `  H
) `  X )
) )
159fveq2d 5529 . . 3  |-  ( (
ph  /\  x  =  X )  ->  ( S `  x )  =  ( S `  X ) )
169fveq2d 5529 . . 3  |-  ( (
ph  /\  x  =  X )  ->  ( R `  x )  =  ( R `  X ) )
1714, 15, 16oveq123d 5879 . 2  |-  ( (
ph  /\  x  =  X )  ->  (
( S `  x
) ( <. (
( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >.  .x.  ( ( 1st `  H ) `  x ) ) ( R `  x ) )  =  ( ( S `  X ) ( <. ( ( 1st `  F ) `  X
) ,  ( ( 1st `  G ) `
 X ) >.  .x.  ( ( 1st `  H
) `  X )
) ( R `  X ) ) )
18 fuccoval.f . 2  |-  ( ph  ->  X  e.  A )
19 ovex 5883 . . 3  |-  ( ( S `  X ) ( <. ( ( 1st `  F ) `  X
) ,  ( ( 1st `  G ) `
 X ) >.  .x.  ( ( 1st `  H
) `  X )
) ( R `  X ) )  e. 
_V
2019a1i 10 . 2  |-  ( ph  ->  ( ( S `  X ) ( <.
( ( 1st `  F
) `  X ) ,  ( ( 1st `  G ) `  X
) >.  .x.  ( ( 1st `  H ) `  X ) ) ( R `  X ) )  e.  _V )
218, 17, 18, 20fvmptd 5606 1  |-  ( ph  ->  ( ( S (
<. F ,  G >.  .xb 
H ) R ) `
 X )  =  ( ( S `  X ) ( <.
( ( 1st `  F
) `  X ) ,  ( ( 1st `  G ) `  X
) >.  .x.  ( ( 1st `  H ) `  X ) ) ( R `  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643   ` cfv 5255  (class class class)co 5858   1stc1st 6120   Basecbs 13148  compcco 13220   Nat cnat 13815   FuncCat cfuc 13816
This theorem is referenced by:  fuccocl  13838  fucass  13842  evlfcllem  13995  yonedalem3b  14053
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-hom 13232  df-cco 13233  df-func 13732  df-nat 13817  df-fuc 13818
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