Theorem subsubc 15091

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 2441	. . . . . 6 |- ( Hom f ` D ) = ( Hom f ` D )
3	1, 2	subcssc 15078	. . . . 5 |- ( J e. (Subcat ` D ) -> J C_cat ( Hom f ` D ) )
4		subsubc.d	. . . . . . 7 |- D = ( C |`cat H )
5		eqid 2441	. . . . . . 7 |- ( Base ` C ) = ( Base ` C )
6		subcrcl 15057	. . . . . . 7 |- ( H e. (Subcat ` C ) -> C e. Cat )
7		id 22	. . . . . . . 8 |- ( H e. (Subcat ` C ) -> H e. (Subcat ` C ) )
8		eqidd 2442	. . . . . . . 8 |- ( H e. (Subcat ` C ) -> dom dom H = dom dom H )
9	7, 8	subcfn 15079	. . . . . . 7 |- ( H e. (Subcat ` C ) -> H Fn ( dom dom H X. dom dom H ) )
10	7, 9, 5	subcss1 15080	. . . . . . 7 |- ( H e. (Subcat ` C ) -> dom dom H C_ ( Base ` C ) )
11	4, 5, 6, 9, 10	reschomf 15072	. . . . . 6 |- ( H e. (Subcat ` C ) -> H = ( Hom f ` D ) )
12	11	breq2d 4445	. . . . 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 2441	. . . . . . . . 9 |- ( Hom f ` C ) = ( Hom f ` C )
18	16, 17	subcssc 15078	. . . . . . . 8 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> H C_cat ( Hom f ` C ) )
19		ssctr 15066	. . . . . . . 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 2441	. . . . . . . . 9 |- ( Id ` C ) = ( Id ` C )
27		eqidd 2442	. . . . . . . . . . . 12 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> dom dom J = dom dom J )
28	15, 27	sscfn1 15058	. . . . . . . . . . 11 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> J Fn ( dom dom J X. dom dom J ) )
29	28, 24, 15	ssc1 15062	. . . . . . . . . 10 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> dom dom J C_ dom dom H )
30	29	sselda 3486	. . . . . . . . 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 15085	. . . . . . . 8 |- ( ( ( H e. (Subcat ` C ) /\ J C_cat H ) /\ x e. dom dom J ) -> ( ( Id ` C ) ` x ) = ( ( Id ` D ) ` x ) )
32	31	eleq1d 2510	. . . . . . 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 2877	. . . . . 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 6287	. . . . . . . 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 6712	. . . . . . . . . . 11 |- ( H e. (Subcat ` C ) -> dom H e. _V )
37		dmexg 6712	. . . . . . . . . . 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 15074	. . . . . . . 8 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> ( ( C |`cat H ) |`cat J ) = ( C |`cat J ) )
41	34, 40	syl5req 2495	. . . . . . 7 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> ( C |`cat J ) = ( D |`cat J ) )
42	41	eleq1d 2510	. . . . . 6 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> ( ( C |`cat J ) e. Cat <-> ( D |`cat J ) e. Cat ) )
43	22, 33, 42	3anbi123d 1298	. . . . 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 2441	. . . . . 6 |- ( C |`cat J ) = ( C |`cat J )
45	17, 26, 44, 35, 28	issubc3 15087	. . . . 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 2441	. . . . . 6 |- ( Id ` D ) = ( Id ` D )
47		eqid 2441	. . . . . 6 |- ( D |`cat J ) = ( D |`cat J )
48	4, 7	subccat 15086	. . . . . . 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 15087	. . . . 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 972   = wceq 1381   e. wcel 1802   A.wral 2791   _Vcvv 3093   class class class wbr 4433   X. cxp 4983   dom cdm 4985   Fn wfn 5569   ` cfv 5574  (class class class)co 6277   Basecbs 14504   Catccat 14933   Idccid 14934   Hom <sub>f</sub> chomf 14935   C_<sub>cat</sub> cssc 15048   |`_cat cresc 15049  Subcatcsubc 15050

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-er 7309  df-pm 7421  df-ixp 7468  df-en 7515  df-dom 7516  df-sdom 7517  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10980  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-ress 14511  df-hom 14593  df-cco 14594  df-cat 14937  df-cid 14938  df-homf 14939  df-ssc 15051  df-resc 15052  df-subc 15053

This theorem is referenced by:  fldhmsubc  32600  fldhmsubcOLD  32619