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Theorem fuccoval 14080
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 14079 . 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 448 . . . . . 6  |-  ( (
ph  /\  x  =  X )  ->  x  =  X )
109fveq2d 5665 . . . . 5  |-  ( (
ph  /\  x  =  X )  ->  (
( 1st `  F
) `  x )  =  ( ( 1st `  F ) `  X
) )
119fveq2d 5665 . . . . 5  |-  ( (
ph  /\  x  =  X )  ->  (
( 1st `  G
) `  x )  =  ( ( 1st `  G ) `  X
) )
1210, 11opeq12d 3927 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  <. (
( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >.  =  <. (
( 1st `  F
) `  X ) ,  ( ( 1st `  G ) `  X
) >. )
139fveq2d 5665 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  (
( 1st `  H
) `  x )  =  ( ( 1st `  H ) `  X
) )
1412, 13oveq12d 6031 . . 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 5665 . . 3  |-  ( (
ph  /\  x  =  X )  ->  ( S `  x )  =  ( S `  X ) )
169fveq2d 5665 . . 3  |-  ( (
ph  /\  x  =  X )  ->  ( R `  x )  =  ( R `  X ) )
1714, 15, 16oveq123d 6034 . 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 6038 . . 3  |-  ( ( S `  X ) ( <. ( ( 1st `  F ) `  X
) ,  ( ( 1st `  G ) `
 X ) >.  .x.  ( ( 1st `  H
) `  X )
) ( R `  X ) )  e. 
_V
2019a1i 11 . 2  |-  ( ph  ->  ( ( S `  X ) ( <.
( ( 1st `  F
) `  X ) ,  ( ( 1st `  G ) `  X
) >.  .x.  ( ( 1st `  H ) `  X ) ) ( R `  X ) )  e.  _V )
218, 17, 18, 20fvmptd 5742 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 359    = wceq 1649    e. wcel 1717   _Vcvv 2892   <.cop 3753   ` cfv 5387  (class class class)co 6013   1stc1st 6279   Basecbs 13389  compcco 13461   Nat cnat 14058   FuncCat cfuc 14059
This theorem is referenced by:  fuccocl  14081  fucass  14085  evlfcllem  14238  yonedalem3b  14296
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-oadd 6657  df-er 6834  df-ixp 6993  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-nn 9926  df-2 9983  df-3 9984  df-4 9985  df-5 9986  df-6 9987  df-7 9988  df-8 9989  df-9 9990  df-10 9991  df-n0 10147  df-z 10208  df-dec 10308  df-uz 10414  df-fz 10969  df-struct 13391  df-ndx 13392  df-slot 13393  df-base 13394  df-hom 13473  df-cco 13474  df-func 13975  df-nat 14060  df-fuc 14061
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