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Theorem catcco 13933
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 )
catcco.x  |-  ( ph  ->  X  e.  B )
catcco.y  |-  ( ph  ->  Y  e.  B )
catcco.z  |-  ( ph  ->  Z  e.  B )
catcco.f  |-  ( ph  ->  F  e.  ( X 
Func  Y ) )
catcco.g  |-  ( ph  ->  G  e.  ( Y 
Func  Z ) )
Assertion
Ref Expression
catcco  |-  ( ph  ->  ( G ( <. X ,  Y >.  .x. 
Z ) F )  =  ( G  o.func  F ) )

Proof of Theorem catcco
Dummy variables  v 
z  f  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 catcbas.c . . . 4  |-  C  =  (CatCat `  U )
2 catcbas.b . . . 4  |-  B  =  ( Base `  C
)
3 catcbas.u . . . 4  |-  ( ph  ->  U  e.  V )
4 catcco.o . . . 4  |-  .x.  =  (comp `  C )
51, 2, 3, 4catccofval 13932 . . 3  |-  ( ph  ->  .x.  =  ( v  e.  ( B  X.  B ) ,  z  e.  B  |->  ( g  e.  ( ( 2nd `  v )  Func  z
) ,  f  e.  (  Func  `  v ) 
|->  ( g  o.func  f )
) ) )
6 simprl 732 . . . . . . 7  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  v  =  <. X ,  Y >. )
76fveq2d 5529 . . . . . 6  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 2nd `  v )  =  ( 2nd `  <. X ,  Y >. )
)
8 catcco.x . . . . . . . 8  |-  ( ph  ->  X  e.  B )
9 catcco.y . . . . . . . 8  |-  ( ph  ->  Y  e.  B )
10 op2ndg 6133 . . . . . . . 8  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
118, 9, 10syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
1211adantr 451 . . . . . 6  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
137, 12eqtrd 2315 . . . . 5  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 2nd `  v )  =  Y )
14 simprr 733 . . . . 5  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  z  =  Z )
1513, 14oveq12d 5876 . . . 4  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (
( 2nd `  v
)  Func  z )  =  ( Y  Func  Z ) )
166fveq2d 5529 . . . . 5  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (  Func  `  v )  =  (  Func  `  <. X ,  Y >. ) )
17 df-ov 5861 . . . . 5  |-  ( X 
Func  Y )  =  ( 
Func  `  <. X ,  Y >. )
1816, 17syl6eqr 2333 . . . 4  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (  Func  `  v )  =  ( X  Func  Y
) )
19 eqidd 2284 . . . 4  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (
g  o.func  f )  =  ( g  o.func  f ) )
2015, 18, 19mpt2eq123dv 5910 . . 3  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (
g  e.  ( ( 2nd `  v ) 
Func  z ) ,  f  e.  (  Func  `  v )  |->  ( g  o.func  f ) )  =  ( g  e.  ( Y  Func  Z ) ,  f  e.  ( X  Func  Y )  |->  ( g  o.func  f ) ) )
21 opelxpi 4721 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  B )  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
228, 9, 21syl2anc 642 . . 3  |-  ( ph  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
23 catcco.z . . 3  |-  ( ph  ->  Z  e.  B )
24 ovex 5883 . . . . 5  |-  ( Y 
Func  Z )  e.  _V
25 ovex 5883 . . . . 5  |-  ( X 
Func  Y )  e.  _V
2624, 25mpt2ex 6198 . . . 4  |-  ( g  e.  ( Y  Func  Z ) ,  f  e.  ( X  Func  Y
)  |->  ( g  o.func  f ) )  e.  _V
2726a1i 10 . . 3  |-  ( ph  ->  ( g  e.  ( Y  Func  Z ) ,  f  e.  ( X  Func  Y )  |->  ( g  o.func  f ) )  e. 
_V )
285, 20, 22, 23, 27ovmpt2d 5975 . 2  |-  ( ph  ->  ( <. X ,  Y >.  .x.  Z )  =  ( g  e.  ( Y  Func  Z ) ,  f  e.  ( X  Func  Y )  |->  ( g  o.func  f ) ) )
29 oveq12 5867 . . 3  |-  ( ( g  =  G  /\  f  =  F )  ->  ( g  o.func  f )  =  ( G  o.func  F ) )
3029adantl 452 . 2  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( g  o.func  f )  =  ( G  o.func  F ) )
31 catcco.g . 2  |-  ( ph  ->  G  e.  ( Y 
Func  Z ) )
32 catcco.f . 2  |-  ( ph  ->  F  e.  ( X 
Func  Y ) )
33 ovex 5883 . . 3  |-  ( G  o.func 
F )  e.  _V
3433a1i 10 . 2  |-  ( ph  ->  ( G  o.func  F )  e.  _V )
3528, 30, 31, 32, 34ovmpt2d 5975 1  |-  ( ph  ->  ( G ( <. X ,  Y >.  .x. 
Z ) F )  =  ( G  o.func  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643    X. cxp 4687   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   2ndc2nd 6121   Basecbs 13148  compcco 13220    Func cfunc 13728    o.func ccofu 13730  CatCatccatc 13926
This theorem is referenced by:  catccatid  13934  resscatc  13937  catcisolem  13938  catciso  13939
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-hom 13232  df-cco 13233  df-catc 13927
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