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Theorem fuccoval 15451
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 15450 . 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 459 . . . . . 6  |-  ( (
ph  /\  x  =  X )  ->  x  =  X )
109fveq2d 5852 . . . . 5  |-  ( (
ph  /\  x  =  X )  ->  (
( 1st `  F
) `  x )  =  ( ( 1st `  F ) `  X
) )
119fveq2d 5852 . . . . 5  |-  ( (
ph  /\  x  =  X )  ->  (
( 1st `  G
) `  x )  =  ( ( 1st `  G ) `  X
) )
1210, 11opeq12d 4211 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  <. (
( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >.  =  <. (
( 1st `  F
) `  X ) ,  ( ( 1st `  G ) `  X
) >. )
139fveq2d 5852 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  (
( 1st `  H
) `  x )  =  ( ( 1st `  H ) `  X
) )
1412, 13oveq12d 6288 . . 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 5852 . . 3  |-  ( (
ph  /\  x  =  X )  ->  ( S `  x )  =  ( S `  X ) )
169fveq2d 5852 . . 3  |-  ( (
ph  /\  x  =  X )  ->  ( R `  x )  =  ( R `  X ) )
1714, 15, 16oveq123d 6291 . 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 6298 . . 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 5936 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 setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   _Vcvv 3106   <.cop 4022   ` cfv 5570  (class class class)co 6270   1stc1st 6771   Basecbs 14716  compcco 14796   Nat cnat 15429   FuncCat cfuc 15430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-fz 11676  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-hom 14808  df-cco 14809  df-func 15346  df-nat 15431  df-fuc 15432
This theorem is referenced by:  fuccocl  15452  fucass  15456  evlfcllem  15689  yonedalem3b  15747
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