MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  catccofval Unicode version

Theorem catccofval 13948
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 13945 . . . 4  |-  ( ph  ->  B  =  ( U  i^i  Cat ) )
5 eqid 2296 . . . . 5  |-  (  Hom  `  C )  =  (  Hom  `  C )
61, 3, 2, 5catchomfval 13946 . . . 4  |-  ( ph  ->  (  Hom  `  C
)  =  ( x  e.  B ,  y  e.  B  |->  ( x 
Func  y ) ) )
7 eqidd 2297 . . . 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 13944 . . 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 5545 . 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 5555 . . . . . 6  |-  ( Base `  C )  e.  _V
123, 11eqeltri 2366 . . . . 5  |-  B  e. 
_V
1312, 12xpex 4817 . . . 4  |-  ( B  X.  B )  e. 
_V
1413, 12mpt2ex 6214 . . 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 13847 . . . 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 df-cco 13249 . . . . 5  |- comp  = Slot ; 1 5
17 1nn0 9997 . . . . . 6  |-  1  e.  NN0
18 5nn 9896 . . . . . 6  |-  5  e.  NN
1917, 18decnncl 10153 . . . . 5  |- ; 1 5  e.  NN
2016, 19ndxid 13185 . . . 4  |- comp  = Slot  (comp ` 
ndx )
21 snsstp3 3784 . . . 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 ) ) )
>. }
2215, 20, 21strfv 13196 . . 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 ) ) )
>. } ) )
2314, 22ax-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 ) ) )
>. } )
249, 10, 233eqtr4g 2353 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 1632    e. wcel 1696   _Vcvv 2801   {ctp 3655   <.cop 3656    X. cxp 4703   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   2ndc2nd 6137   1c1 8754   5c5 9814  ;cdc 10140   ndxcnx 13161   Basecbs 13164    Hom chom 13235  compcco 13236    Func cfunc 13744    o.func ccofu 13746  CatCatccatc 13942
This theorem is referenced by:  catcco  13949
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-hom 13248  df-cco 13249  df-catc 13943
  Copyright terms: Public domain W3C validator