Theorem subccocl 16505

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.

Step	Hyp	Ref	Expression
1		subcidcl.j	. . . 4 |- ( ph -> J e. (Subcat ` C ) )
2		eqid 2622	. . . . 5 |- ( Hom f ` C ) = ( Hom f ` C )
3		eqid 2622	. . . . 5 |- ( Id ` C ) = ( Id ` C )
4		subccocl.o	. . . . 5 |- .x. = (comp ` C )
5		subcrcl 16476	. . . . . 6 |- ( J e. (Subcat ` C ) -> C e. Cat )
6	1, 5	syl 17	. . . . 5 |- ( ph -> C e. Cat )
7		subcidcl.2	. . . . 5 |- ( ph -> J Fn ( S X. S ) )
8	2, 3, 4, 6, 7	issubc2 16496	. . . 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 ) ) ) ) )
9	1, 8	mpbid 222	. . 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 ) ) ) )
10	9	simprd 479	. 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 )
13	12	adantr 481	. . . . 5 |- ( ( ph /\ x = X ) -> Y e. S )
14		subccocl.z	. . . . . . 7 |- ( ph -> Z e. S )
15	14	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ x = X ) /\ y = Y ) -> Z e. S )
16		subccocl.f	. . . . . . . . 9 |- ( ph -> F e. ( X J Y ) )
17	16	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) -> F e. ( X J Y ) )
18		simpllr 799	. . . . . . . . 9 |- ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) -> x = X )
19		simplr 792	. . . . . . . . 9 |- ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) -> y = Y )
20	18, 19	oveq12d 6668	. . . . . . . 8 |- ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) -> ( x J y ) = ( X J Y ) )
21	17, 20	eleqtrrd 2704	. . . . . . 7 |- ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) -> F e. ( x J y ) )
22		subccocl.g	. . . . . . . . . 10 |- ( ph -> G e. ( Y J Z ) )
23	22	ad4antr 768	. . . . . . . . 9 |- ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) -> G e. ( Y J Z ) )
24		simpllr 799	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) -> y = Y )
25		simplr 792	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) -> z = Z )
26	24, 25	oveq12d 6668	. . . . . . . . 9 |- ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) -> ( y J z ) = ( Y J Z ) )
27	23, 26	eleqtrrd 2704	. . . . . . . 8 |- ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) -> G e. ( y J z ) )
28		simp-5r 809	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) /\ g = G ) -> x = X )
29		simp-4r 807	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) /\ g = G ) -> y = Y )
30	28, 29	opeq12d 4410	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) /\ g = G ) -> <. x , y >. = <. X , Y >. )
31		simpllr 799	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) /\ g = G ) -> z = Z )
32	30, 31	oveq12d 6668	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) /\ g = G ) -> ( <. x , y >. .x. z ) = ( <. X , Y >. .x. Z ) )
33		simpr 477	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) /\ g = G ) -> g = G )
34		simplr 792	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) /\ g = G ) -> f = F )
35	32, 33, 34	oveq123d 6671	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) /\ g = G ) -> ( g ( <. x , y >. .x. z ) f ) = ( G ( <. X , Y >. .x. Z ) F ) )
36	28, 31	oveq12d 6668	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) /\ g = G ) -> ( x J z ) = ( X J Z ) )
37	35, 36	eleq12d 2695	. . . . . . . 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 ) ) )
38	27, 37	rspcdv 3312	. . . . . . 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 ) ) )
39	21, 38	rspcimdv 3310	. . . . . 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 ) ) )
40	15, 39	rspcimdv 3310	. . . . 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 ) ) )
41	13, 40	rspcimdv 3310	. . . 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 ) ) )
42	41	adantld 483	. . 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 ) ) )
43	11, 42	rspcimdv 3310	. 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 ) ) )
44	10, 43	mpd 15	1 |- ( ph -> ( G ( <. X , Y >. .x. Z ) F ) e. ( X J Z ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   <.cop 4183   class class class wbr 4653   X. cxp 5112   Fn wfn 5883   ` cfv 5888  (class class class)co 6650  compcco 15953   Catccat 16325   Idccid 16326   Hom _f chomf 16327   C__cat cssc 16467  Subcatcsubc 16469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-pm 7860  df-ixp 7909  df-ssc 16470  df-subc 16472

This theorem is referenced by:  subccatid  16506  funcres  16556