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Theorem subccocl 14747
Description: A subcategory is closed under composition. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
subcidcl.j  |-  ( ph  ->  J  e.  (Subcat `  C ) )
subcidcl.2  |-  ( ph  ->  J  Fn  ( S  X.  S ) )
subcidcl.x  |-  ( ph  ->  X  e.  S )
subccocl.o  |-  .x.  =  (comp `  C )
subccocl.y  |-  ( ph  ->  Y  e.  S )
subccocl.z  |-  ( ph  ->  Z  e.  S )
subccocl.f  |-  ( ph  ->  F  e.  ( X J Y ) )
subccocl.g  |-  ( ph  ->  G  e.  ( Y J Z ) )
Assertion
Ref Expression
subccocl  |-  ( ph  ->  ( G ( <. X ,  Y >.  .x. 
Z ) F )  e.  ( X J Z ) )

Proof of Theorem subccocl
Dummy variables  f 
g  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 subcidcl.j . . . 4  |-  ( ph  ->  J  e.  (Subcat `  C ) )
2 eqid 2438 . . . . 5  |-  ( Hom f  `  C )  =  ( Hom f  `  C )
3 eqid 2438 . . . . 5  |-  ( Id
`  C )  =  ( Id `  C
)
4 subccocl.o . . . . 5  |-  .x.  =  (comp `  C )
5 subcrcl 14721 . . . . . 6  |-  ( J  e.  (Subcat `  C
)  ->  C  e.  Cat )
61, 5syl 16 . . . . 5  |-  ( ph  ->  C  e.  Cat )
7 subcidcl.2 . . . . 5  |-  ( ph  ->  J  Fn  ( S  X.  S ) )
82, 3, 4, 6, 7issubc2 14741 . . . 4  |-  ( ph  ->  ( J  e.  (Subcat `  C )  <->  ( J  C_cat  ( Hom f  `  C )  /\  A. x  e.  S  (
( ( Id `  C ) `  x
)  e.  ( x J x )  /\  A. y  e.  S  A. z  e.  S  A. f  e.  ( x J y ) A. g  e.  ( y J z ) ( g ( <. x ,  y >.  .x.  z
) f )  e.  ( x J z ) ) ) ) )
91, 8mpbid 210 . . 3  |-  ( ph  ->  ( J  C_cat  ( Hom f  `  C )  /\  A. x  e.  S  (
( ( Id `  C ) `  x
)  e.  ( x J x )  /\  A. y  e.  S  A. z  e.  S  A. f  e.  ( x J y ) A. g  e.  ( y J z ) ( g ( <. x ,  y >.  .x.  z
) f )  e.  ( x J z ) ) ) )
109simprd 463 . 2  |-  ( ph  ->  A. x  e.  S  ( ( ( Id
`  C ) `  x )  e.  ( x J x )  /\  A. y  e.  S  A. z  e.  S  A. f  e.  ( x J y ) A. g  e.  ( y J z ) ( g (
<. x ,  y >.  .x.  z ) f )  e.  ( x J z ) ) )
11 subcidcl.x . . 3  |-  ( ph  ->  X  e.  S )
12 subccocl.y . . . . . 6  |-  ( ph  ->  Y  e.  S )
1312adantr 465 . . . . 5  |-  ( (
ph  /\  x  =  X )  ->  Y  e.  S )
14 subccocl.z . . . . . . 7  |-  ( ph  ->  Z  e.  S )
1514ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  Z  e.  S )
16 subccocl.f . . . . . . . . 9  |-  ( ph  ->  F  e.  ( X J Y ) )
1716ad3antrrr 729 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  ->  F  e.  ( X J Y ) )
18 simpllr 758 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  ->  x  =  X )
19 simplr 754 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  ->  y  =  Y )
2018, 19oveq12d 6104 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  ->  (
x J y )  =  ( X J Y ) )
2117, 20eleqtrrd 2515 . . . . . . 7  |-  ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  ->  F  e.  ( x J y ) )
22 subccocl.g . . . . . . . . . 10  |-  ( ph  ->  G  e.  ( Y J Z ) )
2322ad4antr 731 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  ->  G  e.  ( Y J Z ) )
24 simpllr 758 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  ->  y  =  Y )
25 simplr 754 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  ->  z  =  Z )
2624, 25oveq12d 6104 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  ->  (
y J z )  =  ( Y J Z ) )
2723, 26eleqtrrd 2515 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  ->  G  e.  ( y J z ) )
28 simp-5r 768 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  ->  x  =  X )
29 simp-4r 766 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  ->  y  =  Y )
3028, 29opeq12d 4062 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  ->  <. x ,  y >.  =  <. X ,  Y >. )
31 simpllr 758 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  ->  z  =  Z )
3230, 31oveq12d 6104 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  ->  ( <. x ,  y >.  .x.  z )  =  (
<. X ,  Y >.  .x. 
Z ) )
33 simpr 461 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  ->  g  =  G )
34 simplr 754 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  ->  f  =  F )
3532, 33, 34oveq123d 6107 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  ->  (
g ( <. x ,  y >.  .x.  z
) f )  =  ( G ( <. X ,  Y >.  .x. 
Z ) F ) )
3628, 31oveq12d 6104 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  ->  (
x J z )  =  ( X J Z ) )
3735, 36eleq12d 2506 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  ->  (
( g ( <.
x ,  y >.  .x.  z ) f )  e.  ( x J z )  <->  ( G
( <. X ,  Y >.  .x.  Z ) F )  e.  ( X J Z ) ) )
3827, 37rspcdv 3071 . . . . . . 7  |-  ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  ->  ( A. g  e.  (
y J z ) ( g ( <.
x ,  y >.  .x.  z ) f )  e.  ( x J z )  ->  ( G ( <. X ,  Y >.  .x.  Z ) F )  e.  ( X J Z ) ) )
3921, 38rspcimdv 3069 . . . . . 6  |-  ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  ->  ( A. f  e.  (
x J y ) A. g  e.  ( y J z ) ( g ( <.
x ,  y >.  .x.  z ) f )  e.  ( x J z )  ->  ( G ( <. X ,  Y >.  .x.  Z ) F )  e.  ( X J Z ) ) )
4015, 39rspcimdv 3069 . . . . 5  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  ( A. z  e.  S  A. f  e.  (
x J y ) A. g  e.  ( y J z ) ( g ( <.
x ,  y >.  .x.  z ) f )  e.  ( x J z )  ->  ( G ( <. X ,  Y >.  .x.  Z ) F )  e.  ( X J Z ) ) )
4113, 40rspcimdv 3069 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  ( A. y  e.  S  A. z  e.  S  A. f  e.  (
x J y ) A. g  e.  ( y J z ) ( g ( <.
x ,  y >.  .x.  z ) f )  e.  ( x J z )  ->  ( G ( <. X ,  Y >.  .x.  Z ) F )  e.  ( X J Z ) ) )
4241adantld 467 . . 3  |-  ( (
ph  /\  x  =  X )  ->  (
( ( ( Id
`  C ) `  x )  e.  ( x J x )  /\  A. y  e.  S  A. z  e.  S  A. f  e.  ( x J y ) A. g  e.  ( y J z ) ( g (
<. x ,  y >.  .x.  z ) f )  e.  ( x J z ) )  -> 
( G ( <. X ,  Y >.  .x. 
Z ) F )  e.  ( X J Z ) ) )
4311, 42rspcimdv 3069 . 2  |-  ( ph  ->  ( A. x  e.  S  ( ( ( Id `  C ) `
 x )  e.  ( x J x )  /\  A. y  e.  S  A. z  e.  S  A. f  e.  ( x J y ) A. g  e.  ( y J z ) ( g (
<. x ,  y >.  .x.  z ) f )  e.  ( x J z ) )  -> 
( G ( <. X ,  Y >.  .x. 
Z ) F )  e.  ( X J Z ) ) )
4410, 43mpd 15 1  |-  ( ph  ->  ( G ( <. X ,  Y >.  .x. 
Z ) F )  e.  ( X J Z ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2710   <.cop 3878   class class class wbr 4287    X. cxp 4833    Fn wfn 5408   ` cfv 5413  (class class class)co 6086  compcco 14242   Catccat 14594   Idccid 14595   Hom f chomf 14596    C_cat cssc 14712  Subcatcsubc 14714
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
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-fal 1375  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-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-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  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-ov 6089  df-oprab 6090  df-mpt2 6091  df-pm 7209  df-ixp 7256  df-ssc 14715  df-subc 14717
This theorem is referenced by:  subccatid  14748  funcres  14798
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