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Theorem fuccoval 14993
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 14992 . 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 461 . . . . . 6  |-  ( (
ph  /\  x  =  X )  ->  x  =  X )
109fveq2d 5804 . . . . 5  |-  ( (
ph  /\  x  =  X )  ->  (
( 1st `  F
) `  x )  =  ( ( 1st `  F ) `  X
) )
119fveq2d 5804 . . . . 5  |-  ( (
ph  /\  x  =  X )  ->  (
( 1st `  G
) `  x )  =  ( ( 1st `  G ) `  X
) )
1210, 11opeq12d 4176 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  <. (
( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >.  =  <. (
( 1st `  F
) `  X ) ,  ( ( 1st `  G ) `  X
) >. )
139fveq2d 5804 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  (
( 1st `  H
) `  x )  =  ( ( 1st `  H ) `  X
) )
1412, 13oveq12d 6219 . . 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 5804 . . 3  |-  ( (
ph  /\  x  =  X )  ->  ( S `  x )  =  ( S `  X ) )
169fveq2d 5804 . . 3  |-  ( (
ph  /\  x  =  X )  ->  ( R `  x )  =  ( R `  X ) )
1714, 15, 16oveq123d 6222 . 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 6226 . . 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 5889 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 369    = wceq 1370    e. wcel 1758   _Vcvv 3078   <.cop 3992   ` cfv 5527  (class class class)co 6201   1stc1st 6686   Basecbs 14293  compcco 14370   Nat cnat 14971   FuncCat cfuc 14972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-1o 7031  df-oadd 7035  df-er 7212  df-ixp 7375  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-2 10492  df-3 10493  df-4 10494  df-5 10495  df-6 10496  df-7 10497  df-8 10498  df-9 10499  df-10 10500  df-n0 10692  df-z 10759  df-dec 10868  df-uz 10974  df-fz 11556  df-struct 14295  df-ndx 14296  df-slot 14297  df-base 14298  df-hom 14382  df-cco 14383  df-func 14888  df-nat 14973  df-fuc 14974
This theorem is referenced by:  fuccocl  14994  fucass  14998  evlfcllem  15151  yonedalem3b  15209
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