Theorem subccatid 16506

Description: A subcategory is a category. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
subccat.1	|- D = ( C |`cat J )
subccat.j	|- ( ph -> J e. (Subcat ` C ) )
subccatid.1	|- ( ph -> J Fn ( S X. S ) )
subccatid.2	|- .1. = ( Id ` C )

Assertion

Ref	Expression
subccatid	|- ( ph -> ( D e. Cat /\ ( Id ` D ) = ( x e. S |-> ( .1. ` x ) ) ) )

Distinct variable groups:   x, C   x, D   ph, x   x, .1.   x, J   x, S

Proof of Theorem subccatid

Dummy variables f g h w y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		subccat.1	. . 3 |- D = ( C |`cat J )
2		eqid 2622	. . 3 |- ( Base ` C ) = ( Base ` C )
3		subccat.j	. . . 4 |- ( ph -> J e. (Subcat ` C ) )
4		subcrcl 16476	. . . 4 |- ( J e. (Subcat ` C ) -> C e. Cat )
5	3, 4	syl 17	. . 3 |- ( ph -> C e. Cat )
6		subccatid.1	. . 3 |- ( ph -> J Fn ( S X. S ) )
7	3, 6, 2	subcss1 16502	. . 3 |- ( ph -> S C_ ( Base ` C ) )
8	1, 2, 5, 6, 7	rescbas 16489	. 2 |- ( ph -> S = ( Base ` D ) )
9	1, 2, 5, 6, 7	reschom 16490	. 2 |- ( ph -> J = ( Hom ` D ) )
10		eqid 2622	. . 3 |- (comp ` C ) = (comp ` C )
11	1, 2, 5, 6, 7, 10	rescco 16492	. 2 |- ( ph -> (comp ` C ) = (comp ` D ) )
12		ovex 6678	. . . 4 |- ( C |`cat J ) e. _V
13	1, 12	eqeltri 2697	. . 3 |- D e. _V
14	13	a1i 11	. 2 |- ( ph -> D e. _V )
15		biid 251	. 2 |- ( ( ( w e. S /\ x e. S ) /\ ( y e. S /\ z e. S ) /\ ( f e. ( w J x ) /\ g e. ( x J y ) /\ h e. ( y J z ) ) ) <-> ( ( w e. S /\ x e. S ) /\ ( y e. S /\ z e. S ) /\ ( f e. ( w J x ) /\ g e. ( x J y ) /\ h e. ( y J z ) ) ) )
16	3	adantr 481	. . 3 |- ( ( ph /\ x e. S ) -> J e. (Subcat ` C ) )
17	6	adantr 481	. . 3 |- ( ( ph /\ x e. S ) -> J Fn ( S X. S ) )
18		simpr 477	. . 3 |- ( ( ph /\ x e. S ) -> x e. S )
19		subccatid.2	. . 3 |- .1. = ( Id ` C )
20	16, 17, 18, 19	subcidcl 16504	. 2 |- ( ( ph /\ x e. S ) -> ( .1. ` x ) e. ( x J x ) )
21		eqid 2622	. . 3 |- ( Hom ` C ) = ( Hom ` C )
22	5	adantr 481	. . 3 |- ( ( ph /\ ( ( w e. S /\ x e. S ) /\ ( y e. S /\ z e. S ) /\ ( f e. ( w J x ) /\ g e. ( x J y ) /\ h e. ( y J z ) ) ) ) -> C e. Cat )
23	7	adantr 481	. . . 4 |- ( ( ph /\ ( ( w e. S /\ x e. S ) /\ ( y e. S /\ z e. S ) /\ ( f e. ( w J x ) /\ g e. ( x J y ) /\ h e. ( y J z ) ) ) ) -> S C_ ( Base ` C ) )
24		simpr1l 1118	. . . 4 |- ( ( ph /\ ( ( w e. S /\ x e. S ) /\ ( y e. S /\ z e. S ) /\ ( f e. ( w J x ) /\ g e. ( x J y ) /\ h e. ( y J z ) ) ) ) -> w e. S )
25	23, 24	sseldd 3604	. . 3 |- ( ( ph /\ ( ( w e. S /\ x e. S ) /\ ( y e. S /\ z e. S ) /\ ( f e. ( w J x ) /\ g e. ( x J y ) /\ h e. ( y J z ) ) ) ) -> w e. ( Base ` C ) )
26		simpr1r 1119	. . . 4 |- ( ( ph /\ ( ( w e. S /\ x e. S ) /\ ( y e. S /\ z e. S ) /\ ( f e. ( w J x ) /\ g e. ( x J y ) /\ h e. ( y J z ) ) ) ) -> x e. S )
27	23, 26	sseldd 3604	. . 3 |- ( ( ph /\ ( ( w e. S /\ x e. S ) /\ ( y e. S /\ z e. S ) /\ ( f e. ( w J x ) /\ g e. ( x J y ) /\ h e. ( y J z ) ) ) ) -> x e. ( Base ` C ) )
28	3	adantr 481	. . . . 5 |- ( ( ph /\ ( ( w e. S /\ x e. S ) /\ ( y e. S /\ z e. S ) /\ ( f e. ( w J x ) /\ g e. ( x J y ) /\ h e. ( y J z ) ) ) ) -> J e. (Subcat ` C ) )
29	6	adantr 481	. . . . 5 |- ( ( ph /\ ( ( w e. S /\ x e. S ) /\ ( y e. S /\ z e. S ) /\ ( f e. ( w J x ) /\ g e. ( x J y ) /\ h e. ( y J z ) ) ) ) -> J Fn ( S X. S ) )
30	28, 29, 21, 24, 26	subcss2 16503	. . . 4 |- ( ( ph /\ ( ( w e. S /\ x e. S ) /\ ( y e. S /\ z e. S ) /\ ( f e. ( w J x ) /\ g e. ( x J y ) /\ h e. ( y J z ) ) ) ) -> ( w J x ) C_ ( w ( Hom ` C ) x ) )
31		simpr31 1151	. . . 4 |- ( ( ph /\ ( ( w e. S /\ x e. S ) /\ ( y e. S /\ z e. S ) /\ ( f e. ( w J x ) /\ g e. ( x J y ) /\ h e. ( y J z ) ) ) ) -> f e. ( w J x ) )
32	30, 31	sseldd 3604	. . 3 |- ( ( ph /\ ( ( w e. S /\ x e. S ) /\ ( y e. S /\ z e. S ) /\ ( f e. ( w J x ) /\ g e. ( x J y ) /\ h e. ( y J z ) ) ) ) -> f e. ( w ( Hom ` C ) x ) )
33	2, 21, 19, 22, 25, 10, 27, 32	catlid 16344	. 2 |- ( ( ph /\ ( ( w e. S /\ x e. S ) /\ ( y e. S /\ z e. S ) /\ ( f e. ( w J x ) /\ g e. ( x J y ) /\ h e. ( y J z ) ) ) ) -> ( ( .1. ` x ) ( <. w , x >. (comp ` C ) x ) f ) = f )
34		simpr2l 1120	. . . 4 |- ( ( ph /\ ( ( w e. S /\ x e. S ) /\ ( y e. S /\ z e. S ) /\ ( f e. ( w J x ) /\ g e. ( x J y ) /\ h e. ( y J z ) ) ) ) -> y e. S )
35	23, 34	sseldd 3604	. . 3 |- ( ( ph /\ ( ( w e. S /\ x e. S ) /\ ( y e. S /\ z e. S ) /\ ( f e. ( w J x ) /\ g e. ( x J y ) /\ h e. ( y J z ) ) ) ) -> y e. ( Base ` C ) )
36	28, 29, 21, 26, 34	subcss2 16503	. . . 4 |- ( ( ph /\ ( ( w e. S /\ x e. S ) /\ ( y e. S /\ z e. S ) /\ ( f e. ( w J x ) /\ g e. ( x J y ) /\ h e. ( y J z ) ) ) ) -> ( x J y ) C_ ( x ( Hom ` C ) y ) )
37		simpr32 1152	. . . 4 |- ( ( ph /\ ( ( w e. S /\ x e. S ) /\ ( y e. S /\ z e. S ) /\ ( f e. ( w J x ) /\ g e. ( x J y ) /\ h e. ( y J z ) ) ) ) -> g e. ( x J y ) )
38	36, 37	sseldd 3604	. . 3 |- ( ( ph /\ ( ( w e. S /\ x e. S ) /\ ( y e. S /\ z e. S ) /\ ( f e. ( w J x ) /\ g e. ( x J y ) /\ h e. ( y J z ) ) ) ) -> g e. ( x ( Hom ` C ) y ) )
39	2, 21, 19, 22, 27, 10, 35, 38	catrid 16345	. 2 |- ( ( ph /\ ( ( w e. S /\ x e. S ) /\ ( y e. S /\ z e. S ) /\ ( f e. ( w J x ) /\ g e. ( x J y ) /\ h e. ( y J z ) ) ) ) -> ( g ( <. x , x >. (comp ` C ) y ) ( .1. ` x ) ) = g )
40	28, 29, 24, 10, 26, 34, 31, 37	subccocl 16505	. 2 |- ( ( ph /\ ( ( w e. S /\ x e. S ) /\ ( y e. S /\ z e. S ) /\ ( f e. ( w J x ) /\ g e. ( x J y ) /\ h e. ( y J z ) ) ) ) -> ( g ( <. w , x >. (comp ` C ) y ) f ) e. ( w J y ) )
41		simpr2r 1121	. . . 4 |- ( ( ph /\ ( ( w e. S /\ x e. S ) /\ ( y e. S /\ z e. S ) /\ ( f e. ( w J x ) /\ g e. ( x J y ) /\ h e. ( y J z ) ) ) ) -> z e. S )
42	23, 41	sseldd 3604	. . 3 |- ( ( ph /\ ( ( w e. S /\ x e. S ) /\ ( y e. S /\ z e. S ) /\ ( f e. ( w J x ) /\ g e. ( x J y ) /\ h e. ( y J z ) ) ) ) -> z e. ( Base ` C ) )
43	28, 29, 21, 34, 41	subcss2 16503	. . . 4 |- ( ( ph /\ ( ( w e. S /\ x e. S ) /\ ( y e. S /\ z e. S ) /\ ( f e. ( w J x ) /\ g e. ( x J y ) /\ h e. ( y J z ) ) ) ) -> ( y J z ) C_ ( y ( Hom ` C ) z ) )
44		simpr33 1153	. . . 4 |- ( ( ph /\ ( ( w e. S /\ x e. S ) /\ ( y e. S /\ z e. S ) /\ ( f e. ( w J x ) /\ g e. ( x J y ) /\ h e. ( y J z ) ) ) ) -> h e. ( y J z ) )
45	43, 44	sseldd 3604	. . 3 |- ( ( ph /\ ( ( w e. S /\ x e. S ) /\ ( y e. S /\ z e. S ) /\ ( f e. ( w J x ) /\ g e. ( x J y ) /\ h e. ( y J z ) ) ) ) -> h e. ( y ( Hom ` C ) z ) )
46	2, 21, 10, 22, 25, 27, 35, 32, 38, 42, 45	catass 16347	. 2 |- ( ( ph /\ ( ( w e. S /\ x e. S ) /\ ( y e. S /\ z e. S ) /\ ( f e. ( w J x ) /\ g e. ( x J y ) /\ h e. ( y J z ) ) ) ) -> ( ( h ( <. x , y >. (comp ` C ) z ) g ) ( <. w , x >. (comp ` C ) z ) f ) = ( h ( <. w , y >. (comp ` C ) z ) ( g ( <. w , x >. (comp ` C ) y ) f ) ) )
47	8, 9, 11, 14, 15, 20, 33, 39, 40, 46	iscatd2 16342	1 |- ( ph -> ( D e. Cat /\ ( Id ` D ) = ( x e. S |-> ( .1. ` x ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   |-> cmpt 4729   X. cxp 5112   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   Basecbs 15857   Hom chom 15952  compcco 15953   Catccat 16325   Idccid 16326   |`_cat cresc 16468  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  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-hom 15966  df-cco 15967  df-cat 16329  df-cid 16330  df-homf 16331  df-ssc 16470  df-resc 16471  df-subc 16472

This theorem is referenced by:  subcid  16507  subccat  16508