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Theorem catccofval 15274
Description: Composition in the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
catcbas.c  |-  C  =  (CatCat `  U )
catcbas.b  |-  B  =  ( Base `  C
)
catcbas.u  |-  ( ph  ->  U  e.  V )
catcco.o  |-  .x.  =  (comp `  C )
Assertion
Ref Expression
catccofval  |-  ( ph  ->  .x.  =  ( v  e.  ( B  X.  B ) ,  z  e.  B  |->  ( g  e.  ( ( 2nd `  v )  Func  z
) ,  f  e.  (  Func  `  v ) 
|->  ( g  o.func  f )
) ) )
Distinct variable groups:    z, v, B    f, g, v, z,
ph    v, U, z
Allowed substitution hints:    B( f, g)    C( z, v, f, g)    .x. ( z, v, f, g)    U( f, g)    V( z, v, f, g)

Proof of Theorem catccofval
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 catcbas.c . . . 4  |-  C  =  (CatCat `  U )
2 catcbas.u . . . 4  |-  ( ph  ->  U  e.  V )
3 catcbas.b . . . . 5  |-  B  =  ( Base `  C
)
41, 3, 2catcbas 15271 . . . 4  |-  ( ph  ->  B  =  ( U  i^i  Cat ) )
5 eqid 2460 . . . . 5  |-  ( Hom  `  C )  =  ( Hom  `  C )
61, 3, 2, 5catchomfval 15272 . . . 4  |-  ( ph  ->  ( Hom  `  C
)  =  ( x  e.  B ,  y  e.  B  |->  ( x 
Func  y ) ) )
7 eqidd 2461 . . . 4  |-  ( ph  ->  ( v  e.  ( B  X.  B ) ,  z  e.  B  |->  ( g  e.  ( ( 2nd `  v
)  Func  z ) ,  f  e.  (  Func  `  v )  |->  ( g  o.func  f ) ) )  =  ( v  e.  ( B  X.  B
) ,  z  e.  B  |->  ( g  e.  ( ( 2nd `  v
)  Func  z ) ,  f  e.  (  Func  `  v )  |->  ( g  o.func  f ) ) ) )
81, 2, 4, 6, 7catcval 15270 . . 3  |-  ( ph  ->  C  =  { <. (
Base `  ndx ) ,  B >. ,  <. ( Hom  `  ndx ) ,  ( Hom  `  C
) >. ,  <. (comp ` 
ndx ) ,  ( v  e.  ( B  X.  B ) ,  z  e.  B  |->  ( g  e.  ( ( 2nd `  v ) 
Func  z ) ,  f  e.  (  Func  `  v )  |->  ( g  o.func  f ) ) )
>. } )
98fveq2d 5861 . 2  |-  ( ph  ->  (comp `  C )  =  (comp `  { <. ( Base `  ndx ) ,  B >. ,  <. ( Hom  `  ndx ) ,  ( Hom  `  C
) >. ,  <. (comp ` 
ndx ) ,  ( v  e.  ( B  X.  B ) ,  z  e.  B  |->  ( g  e.  ( ( 2nd `  v ) 
Func  z ) ,  f  e.  (  Func  `  v )  |->  ( g  o.func  f ) ) )
>. } ) )
10 catcco.o . 2  |-  .x.  =  (comp `  C )
11 fvex 5867 . . . . . 6  |-  ( Base `  C )  e.  _V
123, 11eqeltri 2544 . . . . 5  |-  B  e. 
_V
1312, 12xpex 6704 . . . 4  |-  ( B  X.  B )  e. 
_V
1413, 12mpt2ex 6850 . . 3  |-  ( v  e.  ( B  X.  B ) ,  z  e.  B  |->  ( g  e.  ( ( 2nd `  v )  Func  z
) ,  f  e.  (  Func  `  v ) 
|->  ( g  o.func  f )
) )  e.  _V
15 catstr 15173 . . . 4  |-  { <. (
Base `  ndx ) ,  B >. ,  <. ( Hom  `  ndx ) ,  ( Hom  `  C
) >. ,  <. (comp ` 
ndx ) ,  ( v  e.  ( B  X.  B ) ,  z  e.  B  |->  ( g  e.  ( ( 2nd `  v ) 
Func  z ) ,  f  e.  (  Func  `  v )  |->  ( g  o.func  f ) ) )
>. } Struct  <. 1 , ; 1 5 >.
16 ccoid 14662 . . . 4  |- comp  = Slot  (comp ` 
ndx )
17 snsstp3 4173 . . . 4  |-  { <. (comp `  ndx ) ,  ( v  e.  ( B  X.  B ) ,  z  e.  B  |->  ( g  e.  ( ( 2nd `  v ) 
Func  z ) ,  f  e.  (  Func  `  v )  |->  ( g  o.func  f ) ) )
>. }  C_  { <. ( Base `  ndx ) ,  B >. ,  <. ( Hom  `  ndx ) ,  ( Hom  `  C
) >. ,  <. (comp ` 
ndx ) ,  ( v  e.  ( B  X.  B ) ,  z  e.  B  |->  ( g  e.  ( ( 2nd `  v ) 
Func  z ) ,  f  e.  (  Func  `  v )  |->  ( g  o.func  f ) ) )
>. }
1815, 16, 17strfv 14513 . . 3  |-  ( ( v  e.  ( B  X.  B ) ,  z  e.  B  |->  ( g  e.  ( ( 2nd `  v ) 
Func  z ) ,  f  e.  (  Func  `  v )  |->  ( g  o.func  f ) ) )  e.  _V  ->  (
v  e.  ( B  X.  B ) ,  z  e.  B  |->  ( g  e.  ( ( 2nd `  v ) 
Func  z ) ,  f  e.  (  Func  `  v )  |->  ( g  o.func  f ) ) )  =  (comp `  { <. ( Base `  ndx ) ,  B >. , 
<. ( Hom  `  ndx ) ,  ( Hom  `  C ) >. ,  <. (comp `  ndx ) ,  ( v  e.  ( B  X.  B ) ,  z  e.  B  |->  ( g  e.  ( ( 2nd `  v ) 
Func  z ) ,  f  e.  (  Func  `  v )  |->  ( g  o.func  f ) ) )
>. } ) )
1914, 18ax-mp 5 . 2  |-  ( v  e.  ( B  X.  B ) ,  z  e.  B  |->  ( g  e.  ( ( 2nd `  v )  Func  z
) ,  f  e.  (  Func  `  v ) 
|->  ( g  o.func  f )
) )  =  (comp `  { <. ( Base `  ndx ) ,  B >. , 
<. ( Hom  `  ndx ) ,  ( Hom  `  C ) >. ,  <. (comp `  ndx ) ,  ( v  e.  ( B  X.  B ) ,  z  e.  B  |->  ( g  e.  ( ( 2nd `  v ) 
Func  z ) ,  f  e.  (  Func  `  v )  |->  ( g  o.func  f ) ) )
>. } )
209, 10, 193eqtr4g 2526 1  |-  ( ph  ->  .x.  =  ( v  e.  ( B  X.  B ) ,  z  e.  B  |->  ( g  e.  ( ( 2nd `  v )  Func  z
) ,  f  e.  (  Func  `  v ) 
|->  ( g  o.func  f )
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762   _Vcvv 3106   {ctp 4024   <.cop 4026    X. cxp 4990   ` cfv 5579  (class class class)co 6275    |-> cmpt2 6277   2ndc2nd 6773   1c1 9482   5c5 10577  ;cdc 10965   ndxcnx 14476   Basecbs 14479   Hom chom 14555  compcco 14556    Func cfunc 15070    o.func ccofu 15072  CatCatccatc 15268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  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 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-fz 11662  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-hom 14568  df-cco 14569  df-catc 15269
This theorem is referenced by:  catcco  15275
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