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Theorem catcco 14961
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 14960 . . 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 755 . . . . . . 7  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  v  =  <. X ,  Y >. )
76fveq2d 5690 . . . . . 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 6585 . . . . . . . 8  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
118, 9, 10syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
1211adantr 465 . . . . . 6  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
137, 12eqtrd 2470 . . . . 5  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 2nd `  v )  =  Y )
14 simprr 756 . . . . 5  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  z  =  Z )
1513, 14oveq12d 6104 . . . 4  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (
( 2nd `  v
)  Func  z )  =  ( Y  Func  Z ) )
166fveq2d 5690 . . . . 5  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (  Func  `  v )  =  (  Func  `  <. X ,  Y >. ) )
17 df-ov 6089 . . . . 5  |-  ( X 
Func  Y )  =  ( 
Func  `  <. X ,  Y >. )
1816, 17syl6eqr 2488 . . . 4  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (  Func  `  v )  =  ( X  Func  Y
) )
19 eqidd 2439 . . . 4  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (
g  o.func  f )  =  ( g  o.func  f ) )
2015, 18, 19mpt2eq123dv 6143 . . 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 4866 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  B )  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
228, 9, 21syl2anc 661 . . 3  |-  ( ph  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
23 catcco.z . . 3  |-  ( ph  ->  Z  e.  B )
24 ovex 6111 . . . . 5  |-  ( Y 
Func  Z )  e.  _V
25 ovex 6111 . . . . 5  |-  ( X 
Func  Y )  e.  _V
2624, 25mpt2ex 6645 . . . 4  |-  ( g  e.  ( Y  Func  Z ) ,  f  e.  ( X  Func  Y
)  |->  ( g  o.func  f ) )  e.  _V
2726a1i 11 . . 3  |-  ( ph  ->  ( g  e.  ( Y  Func  Z ) ,  f  e.  ( X  Func  Y )  |->  ( g  o.func  f ) )  e. 
_V )
285, 20, 22, 23, 27ovmpt2d 6213 . 2  |-  ( ph  ->  ( <. X ,  Y >.  .x.  Z )  =  ( g  e.  ( Y  Func  Z ) ,  f  e.  ( X  Func  Y )  |->  ( g  o.func  f ) ) )
29 oveq12 6095 . . 3  |-  ( ( g  =  G  /\  f  =  F )  ->  ( g  o.func  f )  =  ( G  o.func  F ) )
3029adantl 466 . 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 6111 . . 3  |-  ( G  o.func 
F )  e.  _V
3433a1i 11 . 2  |-  ( ph  ->  ( G  o.func  F )  e.  _V )
3528, 30, 31, 32, 34ovmpt2d 6213 1  |-  ( ph  ->  ( G ( <. X ,  Y >.  .x. 
Z ) F )  =  ( G  o.func  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2967   <.cop 3878    X. cxp 4833   ` cfv 5413  (class class class)co 6086    e. cmpt2 6088   2ndc2nd 6571   Basecbs 14166  compcco 14242    Func cfunc 14756    o.func ccofu 14758  CatCatccatc 14954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-fz 11430  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-hom 14254  df-cco 14255  df-catc 14955
This theorem is referenced by:  catccatid  14962  resscatc  14965  catcisolem  14966  catciso  14967
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