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Theorem catccofval 15079
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 15076 . . . 4  |-  ( ph  ->  B  =  ( U  i^i  Cat ) )
5 eqid 2451 . . . . 5  |-  ( Hom  `  C )  =  ( Hom  `  C )
61, 3, 2, 5catchomfval 15077 . . . 4  |-  ( ph  ->  ( Hom  `  C
)  =  ( x  e.  B ,  y  e.  B  |->  ( x 
Func  y ) ) )
7 eqidd 2452 . . . 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 15075 . . 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 5796 . 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 5802 . . . . . 6  |-  ( Base `  C )  e.  _V
123, 11eqeltri 2535 . . . . 5  |-  B  e. 
_V
1312, 12xpex 6611 . . . 4  |-  ( B  X.  B )  e. 
_V
1413, 12mpt2ex 6753 . . 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 14978 . . . 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 14467 . . . 4  |- comp  = Slot  (comp ` 
ndx )
17 snsstp3 4127 . . . 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 14319 . . 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 2517 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 1370    e. wcel 1758   _Vcvv 3071   {ctp 3982   <.cop 3984    X. cxp 4939   ` cfv 5519  (class class class)co 6193    |-> cmpt2 6195   2ndc2nd 6679   1c1 9387   5c5 10478  ;cdc 10859   ndxcnx 14282   Basecbs 14285   Hom chom 14360  compcco 14361    Func cfunc 14875    o.func ccofu 14877  CatCatccatc 15073
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 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-1o 7023  df-oadd 7027  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-nn 10427  df-2 10484  df-3 10485  df-4 10486  df-5 10487  df-6 10488  df-7 10489  df-8 10490  df-9 10491  df-10 10492  df-n0 10684  df-z 10751  df-dec 10860  df-uz 10966  df-fz 11548  df-struct 14287  df-ndx 14288  df-slot 14289  df-base 14290  df-hom 14373  df-cco 14374  df-catc 15074
This theorem is referenced by:  catcco  15080
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