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Theorem fuccoval 14152
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 14151 . 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 5724 . . . . 5  |-  ( (
ph  /\  x  =  X )  ->  (
( 1st `  F
) `  x )  =  ( ( 1st `  F ) `  X
) )
119fveq2d 5724 . . . . 5  |-  ( (
ph  /\  x  =  X )  ->  (
( 1st `  G
) `  x )  =  ( ( 1st `  G ) `  X
) )
1210, 11opeq12d 3984 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  <. (
( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >.  =  <. (
( 1st `  F
) `  X ) ,  ( ( 1st `  G ) `  X
) >. )
139fveq2d 5724 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  (
( 1st `  H
) `  x )  =  ( ( 1st `  H ) `  X
) )
1412, 13oveq12d 6091 . . 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 5724 . . 3  |-  ( (
ph  /\  x  =  X )  ->  ( S `  x )  =  ( S `  X ) )
169fveq2d 5724 . . 3  |-  ( (
ph  /\  x  =  X )  ->  ( R `  x )  =  ( R `  X ) )
1714, 15, 16oveq123d 6094 . 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 6098 . . 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 5802 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 1652    e. wcel 1725   _Vcvv 2948   <.cop 3809   ` cfv 5446  (class class class)co 6073   1stc1st 6339   Basecbs 13461  compcco 13533   Nat cnat 14130   FuncCat cfuc 14131
This theorem is referenced by:  fuccocl  14153  fucass  14157  evlfcllem  14310  yonedalem3b  14368
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-fz 11036  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-hom 13545  df-cco 13546  df-func 14047  df-nat 14132  df-fuc 14133
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