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Theorem catccofval 14247
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 14244 . . . 4  |-  ( ph  ->  B  =  ( U  i^i  Cat ) )
5 eqid 2435 . . . . 5  |-  (  Hom  `  C )  =  (  Hom  `  C )
61, 3, 2, 5catchomfval 14245 . . . 4  |-  ( ph  ->  (  Hom  `  C
)  =  ( x  e.  B ,  y  e.  B  |->  ( x 
Func  y ) ) )
7 eqidd 2436 . . . 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 14243 . . 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 5724 . 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 5734 . . . . . 6  |-  ( Base `  C )  e.  _V
123, 11eqeltri 2505 . . . . 5  |-  B  e. 
_V
1312, 12xpex 4982 . . . 4  |-  ( B  X.  B )  e. 
_V
1413, 12mpt2ex 6417 . . 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 14146 . . . 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 13637 . . . 4  |- comp  = Slot  (comp ` 
ndx )
17 snsstp3 3943 . . . 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 13493 . . 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 8 . 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 2492 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 set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   _Vcvv 2948   {ctp 3808   <.cop 3809    X. cxp 4868   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   2ndc2nd 6340   1c1 8983   5c5 10044  ;cdc 10374   ndxcnx 13458   Basecbs 13461    Hom chom 13532  compcco 13533    Func cfunc 14043    o.func ccofu 14045  CatCatccatc 14241
This theorem is referenced by:  catcco  14248
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-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-catc 14242
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