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Theorem catcco 14256
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 14255 . . 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 733 . . . . . . 7  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  v  =  <. X ,  Y >. )
76fveq2d 5732 . . . . . 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 6360 . . . . . . . 8  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
118, 9, 10syl2anc 643 . . . . . . 7  |-  ( ph  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
1211adantr 452 . . . . . 6  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
137, 12eqtrd 2468 . . . . 5  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 2nd `  v )  =  Y )
14 simprr 734 . . . . 5  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  z  =  Z )
1513, 14oveq12d 6099 . . . 4  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (
( 2nd `  v
)  Func  z )  =  ( Y  Func  Z ) )
166fveq2d 5732 . . . . 5  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (  Func  `  v )  =  (  Func  `  <. X ,  Y >. ) )
17 df-ov 6084 . . . . 5  |-  ( X 
Func  Y )  =  ( 
Func  `  <. X ,  Y >. )
1816, 17syl6eqr 2486 . . . 4  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (  Func  `  v )  =  ( X  Func  Y
) )
19 eqidd 2437 . . . 4  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (
g  o.func  f )  =  ( g  o.func  f ) )
2015, 18, 19mpt2eq123dv 6136 . . 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 4910 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  B )  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
228, 9, 21syl2anc 643 . . 3  |-  ( ph  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
23 catcco.z . . 3  |-  ( ph  ->  Z  e.  B )
24 ovex 6106 . . . . 5  |-  ( Y 
Func  Z )  e.  _V
25 ovex 6106 . . . . 5  |-  ( X 
Func  Y )  e.  _V
2624, 25mpt2ex 6425 . . . 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 6201 . 2  |-  ( ph  ->  ( <. X ,  Y >.  .x.  Z )  =  ( g  e.  ( Y  Func  Z ) ,  f  e.  ( X  Func  Y )  |->  ( g  o.func  f ) ) )
29 oveq12 6090 . . 3  |-  ( ( g  =  G  /\  f  =  F )  ->  ( g  o.func  f )  =  ( G  o.func  F ) )
3029adantl 453 . 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 6106 . . 3  |-  ( G  o.func 
F )  e.  _V
3433a1i 11 . 2  |-  ( ph  ->  ( G  o.func  F )  e.  _V )
3528, 30, 31, 32, 34ovmpt2d 6201 1  |-  ( ph  ->  ( G ( <. X ,  Y >.  .x. 
Z ) F )  =  ( G  o.func  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956   <.cop 3817    X. cxp 4876   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   2ndc2nd 6348   Basecbs 13469  compcco 13541    Func cfunc 14051    o.func ccofu 14053  CatCatccatc 14249
This theorem is referenced by:  catccatid  14257  resscatc  14260  catcisolem  14261  catciso  14262
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-fz 11044  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-hom 13553  df-cco 13554  df-catc 14250
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