Theorem subsubc 15751

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 2450	. . . . . 6 |- ( Hom f ` D ) = ( Hom f ` D )
3	1, 2	subcssc 15738	. . . . 5 |- ( J e. (Subcat ` D ) -> J C_cat ( Hom f ` D ) )
4		subsubc.d	. . . . . . 7 |- D = ( C |`cat H )
5		eqid 2450	. . . . . . 7 |- ( Base ` C ) = ( Base ` C )
6		subcrcl 15714	. . . . . . 7 |- ( H e. (Subcat ` C ) -> C e. Cat )
7		id 22	. . . . . . . 8 |- ( H e. (Subcat ` C ) -> H e. (Subcat ` C ) )
8		eqidd 2451	. . . . . . . 8 |- ( H e. (Subcat ` C ) -> dom dom H = dom dom H )
9	7, 8	subcfn 15739	. . . . . . 7 |- ( H e. (Subcat ` C ) -> H Fn ( dom dom H X. dom dom H ) )
10	7, 9, 5	subcss1 15740	. . . . . . 7 |- ( H e. (Subcat ` C ) -> dom dom H C_ ( Base ` C ) )
11	4, 5, 6, 9, 10	reschomf 15729	. . . . . 6 |- ( H e. (Subcat ` C ) -> H = ( Hom f ` D ) )
12	11	breq2d 4413	. . . . 5 |- ( H e. (Subcat ` C ) -> ( J C_cat H <-> J C_cat ( Hom f ` D ) ) )
13	3, 12	syl5ibr 225	. . . 4 |- ( H e. (Subcat ` C ) -> ( J e. (Subcat ` D ) -> J C_cat H ) )
14	13	pm4.71rd 640	. . 3 |- ( H e. (Subcat ` C ) -> ( J e. (Subcat ` D ) <-> ( J C_cat H /\ J e. (Subcat ` D ) ) ) )
15		simpr 463	. . . . . . . 8 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> J C_cat H )
16		simpl 459	. . . . . . . . 9 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> H e. (Subcat ` C ) )
17		eqid 2450	. . . . . . . . 9 |- ( Hom f ` C ) = ( Hom f ` C )
18	16, 17	subcssc 15738	. . . . . . . 8 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> H C_cat ( Hom f ` C ) )
19		ssctr 15723	. . . . . . . 8 |- ( ( J C_cat H /\ H C_cat ( Hom f ` C ) ) -> J C_cat ( Hom f ` C ) )
20	15, 18, 19	syl2anc 666	. . . . . . 7 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> J C_cat ( Hom f ` C ) )
21	12	biimpa 487	. . . . . . 7 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> J C_cat ( Hom f ` D ) )
22	20, 21	2thd 244	. . . . . 6 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> ( J C_cat ( Hom f ` C ) <-> J C_cat ( Hom f ` D ) ) )
23	16	adantr 467	. . . . . . . . 9 |- ( ( ( H e. (Subcat ` C ) /\ J C_cat H ) /\ x e. dom dom J ) -> H e. (Subcat ` C ) )
24	9	adantr 467	. . . . . . . . . 10 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> H Fn ( dom dom H X. dom dom H ) )
25	24	adantr 467	. . . . . . . . 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 2450	. . . . . . . . 9 |- ( Id ` C ) = ( Id ` C )
27		eqidd 2451	. . . . . . . . . . . 12 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> dom dom J = dom dom J )
28	15, 27	sscfn1 15715	. . . . . . . . . . 11 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> J Fn ( dom dom J X. dom dom J ) )
29	28, 24, 15	ssc1 15719	. . . . . . . . . 10 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> dom dom J C_ dom dom H )
30	29	sselda 3431	. . . . . . . . 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 15745	. . . . . . . 8 |- ( ( ( H e. (Subcat ` C ) /\ J C_cat H ) /\ x e. dom dom J ) -> ( ( Id ` C ) ` x ) = ( ( Id ` D ) ` x ) )
32	31	eleq1d 2512	. . . . . . 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 2823	. . . . . 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 6298	. . . . . . . 8 |- ( D |`cat J ) = ( ( C |`cat H ) |`cat J )
35	6	adantr 467	. . . . . . . . 9 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> C e. Cat )
36		dmexg 6721	. . . . . . . . . . 11 |- ( H e. (Subcat ` C ) -> dom H e. _V )
37		dmexg 6721	. . . . . . . . . . 11 |- ( dom H e. _V -> dom dom H e. _V )
38	36, 37	syl 17	. . . . . . . . . 10 |- ( H e. (Subcat ` C ) -> dom dom H e. _V )
39	38	adantr 467	. . . . . . . . 9 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> dom dom H e. _V )
40	35, 24, 28, 39, 29	rescabs 15731	. . . . . . . 8 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> ( ( C |`cat H ) |`cat J ) = ( C |`cat J ) )
41	34, 40	syl5req 2497	. . . . . . 7 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> ( C |`cat J ) = ( D |`cat J ) )
42	41	eleq1d 2512	. . . . . 6 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> ( ( C |`cat J ) e. Cat <-> ( D |`cat J ) e. Cat ) )
43	22, 33, 42	3anbi123d 1338	. . . . 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 2450	. . . . . 6 |- ( C |`cat J ) = ( C |`cat J )
45	17, 26, 44, 35, 28	issubc3 15747	. . . . 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 2450	. . . . . 6 |- ( Id ` D ) = ( Id ` D )
47		eqid 2450	. . . . . 6 |- ( D |`cat J ) = ( D |`cat J )
48	4, 7	subccat 15746	. . . . . . 7 |- ( H e. (Subcat ` C ) -> D e. Cat )
49	48	adantr 467	. . . . . 6 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> D e. Cat )
50	2, 46, 47, 49, 28	issubc3 15747	. . . . 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 290	. . . 4 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> ( J e. (Subcat ` D ) <-> J e. (Subcat ` C ) ) )
52	51	pm5.32da 646	. . 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 257	. 2 |- ( H e. (Subcat ` C ) -> ( J e. (Subcat ` D ) <-> ( J C_cat H /\ J e. (Subcat ` C ) ) ) )
54		ancom 452	. 2 |- ( ( J C_cat H /\ J e. (Subcat ` C ) ) <-> ( J e. (Subcat ` C ) /\ J C_cat H ) )
55	53, 54	syl6bb 265	1 |- ( H e. (Subcat ` C ) -> ( J e. (Subcat ` D ) <-> ( J e. (Subcat ` C ) /\ J C_cat H ) ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 984   = wceq 1443   e. wcel 1886   A.wral 2736   _Vcvv 3044   class class class wbr 4401   X. cxp 4831   dom cdm 4833   Fn wfn 5576   ` cfv 5581  (class class class)co 6288   Basecbs 15114   Catccat 15563   Idccid 15564   Hom <sub>f</sub> chomf 15565   C_<sub>cat</sub> cssc 15705   |`_cat cresc 15706  Subcatcsubc 15707

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-er 7360  df-pm 7472  df-ixp 7520  df-en 7567  df-dom 7568  df-sdom 7569  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-nn 10607  df-2 10665  df-3 10666  df-4 10667  df-5 10668  df-6 10669  df-7 10670  df-8 10671  df-9 10672  df-10 10673  df-n0 10867  df-z 10935  df-dec 11049  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-hom 15207  df-cco 15208  df-cat 15567  df-cid 15568  df-homf 15569  df-ssc 15708  df-resc 15709  df-subc 15710

This theorem is referenced by:  fldhmsubc  40073  fldhmsubcALTV  40092