Theorem subsubc 14762

Description: A subcategory of a subcategory is a subcategory. (Contributed by Mario Carneiro, 6-Jan-2017.)

Hypothesis

Ref	Expression
subsubc.d	|- D = ( C |`cat H )

Assertion

Ref	Expression
subsubc	|- ( H e. (Subcat ` C ) -> ( J e. (Subcat ` D ) <-> ( J e. (Subcat ` C ) /\ J C_cat H ) ) )

Proof of Theorem subsubc

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . . . . 6 |- ( J e. (Subcat ` D ) -> J e. (Subcat ` D ) )
2		eqid 2442	. . . . . 6 |- ( Hom f ` D ) = ( Hom f ` D )
3	1, 2	subcssc 14749	. . . . 5 |- ( J e. (Subcat ` D ) -> J C_cat ( Hom f ` D ) )
4		subsubc.d	. . . . . . 7 |- D = ( C |`cat H )
5		eqid 2442	. . . . . . 7 |- ( Base ` C ) = ( Base ` C )
6		subcrcl 14728	. . . . . . 7 |- ( H e. (Subcat ` C ) -> C e. Cat )
7		id 22	. . . . . . . 8 |- ( H e. (Subcat ` C ) -> H e. (Subcat ` C ) )
8		eqidd 2443	. . . . . . . 8 |- ( H e. (Subcat ` C ) -> dom dom H = dom dom H )
9	7, 8	subcfn 14750	. . . . . . 7 |- ( H e. (Subcat ` C ) -> H Fn ( dom dom H X. dom dom H ) )
10	7, 9, 5	subcss1 14751	. . . . . . 7 |- ( H e. (Subcat ` C ) -> dom dom H C_ ( Base ` C ) )
11	4, 5, 6, 9, 10	reschomf 14743	. . . . . 6 |- ( H e. (Subcat ` C ) -> H = ( Hom f ` D ) )
12	11	breq2d 4303	. . . . 5 |- ( H e. (Subcat ` C ) -> ( J C_cat H <-> J C_cat ( Hom f ` D ) ) )
13	3, 12	syl5ibr 221	. . . 4 |- ( H e. (Subcat ` C ) -> ( J e. (Subcat ` D ) -> J C_cat H ) )
14	13	pm4.71rd 635	. . 3 |- ( H e. (Subcat ` C ) -> ( J e. (Subcat ` D ) <-> ( J C_cat H /\ J e. (Subcat ` D ) ) ) )
15		simpr 461	. . . . . . . 8 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> J C_cat H )
16		simpl 457	. . . . . . . . 9 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> H e. (Subcat ` C ) )
17		eqid 2442	. . . . . . . . 9 |- ( Hom f ` C ) = ( Hom f ` C )
18	16, 17	subcssc 14749	. . . . . . . 8 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> H C_cat ( Hom f ` C ) )
19		ssctr 14737	. . . . . . . 8 |- ( ( J C_cat H /\ H C_cat ( Hom f ` C ) ) -> J C_cat ( Hom f ` C ) )
20	15, 18, 19	syl2anc 661	. . . . . . 7 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> J C_cat ( Hom f ` C ) )
21	12	biimpa 484	. . . . . . 7 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> J C_cat ( Hom f ` D ) )
22	20, 21	2thd 240	. . . . . 6 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> ( J C_cat ( Hom f ` C ) <-> J C_cat ( Hom f ` D ) ) )
23	16	adantr 465	. . . . . . . . 9 |- ( ( ( H e. (Subcat ` C ) /\ J C_cat H ) /\ x e. dom dom J ) -> H e. (Subcat ` C ) )
24	9	adantr 465	. . . . . . . . . 10 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> H Fn ( dom dom H X. dom dom H ) )
25	24	adantr 465	. . . . . . . . 9 |- ( ( ( H e. (Subcat ` C ) /\ J C_cat H ) /\ x e. dom dom J ) -> H Fn ( dom dom H X. dom dom H ) )
26		eqid 2442	. . . . . . . . 9 |- ( Id ` C ) = ( Id ` C )
27		eqidd 2443	. . . . . . . . . . . 12 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> dom dom J = dom dom J )
28	15, 27	sscfn1 14729	. . . . . . . . . . 11 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> J Fn ( dom dom J X. dom dom J ) )
29	28, 24, 15	ssc1 14733	. . . . . . . . . 10 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> dom dom J C_ dom dom H )
30	29	sselda 3355	. . . . . . . . 9 |- ( ( ( H e. (Subcat ` C ) /\ J C_cat H ) /\ x e. dom dom J ) -> x e. dom dom H )
31	4, 23, 25, 26, 30	subcid 14756	. . . . . . . 8 |- ( ( ( H e. (Subcat ` C ) /\ J C_cat H ) /\ x e. dom dom J ) -> ( ( Id ` C ) ` x ) = ( ( Id ` D ) ` x ) )
32	31	eleq1d 2508	. . . . . . 7 |- ( ( ( H e. (Subcat ` C ) /\ J C_cat H ) /\ x e. dom dom J ) -> ( ( ( Id ` C ) ` x ) e. ( x J x ) <-> ( ( Id ` D ) ` x ) e. ( x J x ) ) )
33	32	ralbidva 2730	. . . . . 6 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> ( A. x e. dom dom J ( ( Id ` C ) ` x ) e. ( x J x ) <-> A. x e. dom dom J ( ( Id ` D ) ` x ) e. ( x J x ) ) )
34	4	oveq1i 6100	. . . . . . . 8 |- ( D |`cat J ) = ( ( C |`cat H ) |`cat J )
35	6	adantr 465	. . . . . . . . 9 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> C e. Cat )
36		dmexg 6508	. . . . . . . . . . 11 |- ( H e. (Subcat ` C ) -> dom H e. _V )
37		dmexg 6508	. . . . . . . . . . 11 |- ( dom H e. _V -> dom dom H e. _V )
38	36, 37	syl 16	. . . . . . . . . 10 |- ( H e. (Subcat ` C ) -> dom dom H e. _V )
39	38	adantr 465	. . . . . . . . 9 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> dom dom H e. _V )
40	35, 24, 28, 39, 29	rescabs 14745	. . . . . . . 8 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> ( ( C |`cat H ) |`cat J ) = ( C |`cat J ) )
41	34, 40	syl5req 2487	. . . . . . 7 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> ( C |`cat J ) = ( D |`cat J ) )
42	41	eleq1d 2508	. . . . . 6 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> ( ( C |`cat J ) e. Cat <-> ( D |`cat J ) e. Cat ) )
43	22, 33, 42	3anbi123d 1289	. . . . 5 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> ( ( J C_cat ( Hom f ` C ) /\ A. x e. dom dom J ( ( Id ` C ) ` x ) e. ( x J x ) /\ ( C |`cat J ) e. Cat ) <-> ( J C_cat ( Hom f ` D ) /\ A. x e. dom dom J ( ( Id ` D ) ` x ) e. ( x J x ) /\ ( D |`cat J ) e. Cat ) ) )
44		eqid 2442	. . . . . 6 |- ( C |`cat J ) = ( C |`cat J )
45	17, 26, 44, 35, 28	issubc3 14758	. . . . 5 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> ( J e. (Subcat ` C ) <-> ( J C_cat ( Hom f ` C ) /\ A. x e. dom dom J ( ( Id ` C ) ` x ) e. ( x J x ) /\ ( C |`cat J ) e. Cat ) ) )
46		eqid 2442	. . . . . 6 |- ( Id ` D ) = ( Id ` D )
47		eqid 2442	. . . . . 6 |- ( D |`cat J ) = ( D |`cat J )
48	4, 7	subccat 14757	. . . . . . 7 |- ( H e. (Subcat ` C ) -> D e. Cat )
49	48	adantr 465	. . . . . 6 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> D e. Cat )
50	2, 46, 47, 49, 28	issubc3 14758	. . . . 5 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> ( J e. (Subcat ` D ) <-> ( J C_cat ( Hom f ` D ) /\ A. x e. dom dom J ( ( Id ` D ) ` x ) e. ( x J x ) /\ ( D |`cat J ) e. Cat ) ) )
51	43, 45, 50	3bitr4rd 286	. . . 4 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> ( J e. (Subcat ` D ) <-> J e. (Subcat ` C ) ) )
52	51	pm5.32da 641	. . 3 |- ( H e. (Subcat ` C ) -> ( ( J C_cat H /\ J e. (Subcat ` D ) ) <-> ( J C_cat H /\ J e. (Subcat ` C ) ) ) )
53	14, 52	bitrd 253	. 2 |- ( H e. (Subcat ` C ) -> ( J e. (Subcat ` D ) <-> ( J C_cat H /\ J e. (Subcat ` C ) ) ) )
54		ancom 450	. 2 |- ( ( J C_cat H /\ J e. (Subcat ` C ) ) <-> ( J e. (Subcat ` C ) /\ J C_cat H ) )
55	53, 54	syl6bb 261	1 |- ( H e. (Subcat ` C ) -> ( J e. (Subcat ` D ) <-> ( J e. (Subcat ` C ) /\ J C_cat H ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1369   e. wcel 1756   A.wral 2714   _Vcvv 2971   class class class wbr 4291   X. cxp 4837   dom cdm 4839   Fn wfn 5412   ` cfv 5417  (class class class)co 6090   Basecbs 14173   Catccat 14601   Idccid 14602   Hom <sub>f</sub> chomf 14603   C_<sub>cat</sub> cssc 14719   |`_cat cresc 14720  Subcatcsubc 14721

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 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6831  df-rdg 6865  df-er 7100  df-pm 7216  df-ixp 7263  df-en 7310  df-dom 7311  df-sdom 7312  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-nn 10322  df-2 10379  df-3 10380  df-4 10381  df-5 10382  df-6 10383  df-7 10384  df-8 10385  df-9 10386  df-10 10387  df-n0 10579  df-z 10646  df-dec 10755  df-ndx 14176  df-slot 14177  df-base 14178  df-sets 14179  df-ress 14180  df-hom 14261  df-cco 14262  df-cat 14605  df-cid 14606  df-homf 14607  df-ssc 14722  df-resc 14723  df-subc 14724

This theorem is referenced by: (None)