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Theorem catcco 15092
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 15091 . . 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 5806 . . . . . 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 6703 . . . . . . . 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 2495 . . . . 5  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 2nd `  v )  =  Y )
14 simprr 756 . . . . 5  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  z  =  Z )
1513, 14oveq12d 6221 . . . 4  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (
( 2nd `  v
)  Func  z )  =  ( Y  Func  Z ) )
166fveq2d 5806 . . . . 5  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (  Func  `  v )  =  (  Func  `  <. X ,  Y >. ) )
17 df-ov 6206 . . . . 5  |-  ( X 
Func  Y )  =  ( 
Func  `  <. X ,  Y >. )
1816, 17syl6eqr 2513 . . . 4  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (  Func  `  v )  =  ( X  Func  Y
) )
19 eqidd 2455 . . . 4  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (
g  o.func  f )  =  ( g  o.func  f ) )
2015, 18, 19mpt2eq123dv 6260 . . 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 4982 . . . 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 6228 . . . . 5  |-  ( Y 
Func  Z )  e.  _V
25 ovex 6228 . . . . 5  |-  ( X 
Func  Y )  e.  _V
2624, 25mpt2ex 6763 . . . 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 6331 . 2  |-  ( ph  ->  ( <. X ,  Y >.  .x.  Z )  =  ( g  e.  ( Y  Func  Z ) ,  f  e.  ( X  Func  Y )  |->  ( g  o.func  f ) ) )
29 oveq12 6212 . . 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 6228 . . 3  |-  ( G  o.func 
F )  e.  _V
3433a1i 11 . 2  |-  ( ph  ->  ( G  o.func  F )  e.  _V )
3528, 30, 31, 32, 34ovmpt2d 6331 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 1370    e. wcel 1758   _Vcvv 3078   <.cop 3994    X. cxp 4949   ` cfv 5529  (class class class)co 6203    |-> cmpt2 6205   2ndc2nd 6689   Basecbs 14296  compcco 14373    Func cfunc 14887    o.func ccofu 14889  CatCatccatc 15085
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-nn 10438  df-2 10495  df-3 10496  df-4 10497  df-5 10498  df-6 10499  df-7 10500  df-8 10501  df-9 10502  df-10 10503  df-n0 10695  df-z 10762  df-dec 10871  df-uz 10977  df-fz 11559  df-struct 14298  df-ndx 14299  df-slot 14300  df-base 14301  df-hom 14385  df-cco 14386  df-catc 15086
This theorem is referenced by:  catccatid  15093  resscatc  15096  catcisolem  15097  catciso  15098
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