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Theorem catcco 13949
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 13948 . . 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 5545 . . . . . 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 6149 . . . . . . . 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 2328 . . . . 5  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 2nd `  v )  =  Y )
14 simprr 733 . . . . 5  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  z  =  Z )
1513, 14oveq12d 5892 . . . 4  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (
( 2nd `  v
)  Func  z )  =  ( Y  Func  Z ) )
166fveq2d 5545 . . . . 5  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (  Func  `  v )  =  (  Func  `  <. X ,  Y >. ) )
17 df-ov 5877 . . . . 5  |-  ( X 
Func  Y )  =  ( 
Func  `  <. X ,  Y >. )
1816, 17syl6eqr 2346 . . . 4  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (  Func  `  v )  =  ( X  Func  Y
) )
19 eqidd 2297 . . . 4  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (
g  o.func  f )  =  ( g  o.func  f ) )
2015, 18, 19mpt2eq123dv 5926 . . 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 4737 . . . 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 5899 . . . . 5  |-  ( Y 
Func  Z )  e.  _V
25 ovex 5899 . . . . 5  |-  ( X 
Func  Y )  e.  _V
2624, 25mpt2ex 6214 . . . 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 5991 . 2  |-  ( ph  ->  ( <. X ,  Y >.  .x.  Z )  =  ( g  e.  ( Y  Func  Z ) ,  f  e.  ( X  Func  Y )  |->  ( g  o.func  f ) ) )
29 oveq12 5883 . . 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 5899 . . 3  |-  ( G  o.func 
F )  e.  _V
3433a1i 10 . 2  |-  ( ph  ->  ( G  o.func  F )  e.  _V )
3528, 30, 31, 32, 34ovmpt2d 5991 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 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656    X. cxp 4703   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   2ndc2nd 6137   Basecbs 13164  compcco 13236    Func cfunc 13744    o.func ccofu 13746  CatCatccatc 13942
This theorem is referenced by:  catccatid  13950  resscatc  13953  catcisolem  13954  catciso  13955
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
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