Theorem subsubc 15836

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 2471	. . . . . 6 |- ( Hom f ` D ) = ( Hom f ` D )
3	1, 2	subcssc 15823	. . . . 5 |- ( J e. (Subcat ` D ) -> J C_cat ( Hom f ` D ) )
4		subsubc.d	. . . . . . 7 |- D = ( C |`cat H )
5		eqid 2471	. . . . . . 7 |- ( Base ` C ) = ( Base ` C )
6		subcrcl 15799	. . . . . . 7 |- ( H e. (Subcat ` C ) -> C e. Cat )
7		id 22	. . . . . . . 8 |- ( H e. (Subcat ` C ) -> H e. (Subcat ` C ) )
8		eqidd 2472	. . . . . . . 8 |- ( H e. (Subcat ` C ) -> dom dom H = dom dom H )
9	7, 8	subcfn 15824	. . . . . . 7 |- ( H e. (Subcat ` C ) -> H Fn ( dom dom H X. dom dom H ) )
10	7, 9, 5	subcss1 15825	. . . . . . 7 |- ( H e. (Subcat ` C ) -> dom dom H C_ ( Base ` C ) )
11	4, 5, 6, 9, 10	reschomf 15814	. . . . . 6 |- ( H e. (Subcat ` C ) -> H = ( Hom f ` D ) )
12	11	breq2d 4407	. . . . 5 |- ( H e. (Subcat ` C ) -> ( J C_cat H <-> J C_cat ( Hom f ` D ) ) )
13	3, 12	syl5ibr 229	. . . 4 |- ( H e. (Subcat ` C ) -> ( J e. (Subcat ` D ) -> J C_cat H ) )
14	13	pm4.71rd 647	. . 3 |- ( H e. (Subcat ` C ) -> ( J e. (Subcat ` D ) <-> ( J C_cat H /\ J e. (Subcat ` D ) ) ) )
15		simpr 468	. . . . . . . 8 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> J C_cat H )
16		simpl 464	. . . . . . . . 9 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> H e. (Subcat ` C ) )
17		eqid 2471	. . . . . . . . 9 |- ( Hom f ` C ) = ( Hom f ` C )
18	16, 17	subcssc 15823	. . . . . . . 8 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> H C_cat ( Hom f ` C ) )
19		ssctr 15808	. . . . . . . 8 |- ( ( J C_cat H /\ H C_cat ( Hom f ` C ) ) -> J C_cat ( Hom f ` C ) )
20	15, 18, 19	syl2anc 673	. . . . . . 7 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> J C_cat ( Hom f ` C ) )
21	12	biimpa 492	. . . . . . 7 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> J C_cat ( Hom f ` D ) )
22	20, 21	2thd 248	. . . . . 6 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> ( J C_cat ( Hom f ` C ) <-> J C_cat ( Hom f ` D ) ) )
23	16	adantr 472	. . . . . . . . 9 |- ( ( ( H e. (Subcat ` C ) /\ J C_cat H ) /\ x e. dom dom J ) -> H e. (Subcat ` C ) )
24	9	adantr 472	. . . . . . . . . 10 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> H Fn ( dom dom H X. dom dom H ) )
25	24	adantr 472	. . . . . . . . 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 2471	. . . . . . . . 9 |- ( Id ` C ) = ( Id ` C )
27		eqidd 2472	. . . . . . . . . . . 12 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> dom dom J = dom dom J )
28	15, 27	sscfn1 15800	. . . . . . . . . . 11 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> J Fn ( dom dom J X. dom dom J ) )
29	28, 24, 15	ssc1 15804	. . . . . . . . . 10 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> dom dom J C_ dom dom H )
30	29	sselda 3418	. . . . . . . . 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 15830	. . . . . . . 8 |- ( ( ( H e. (Subcat ` C ) /\ J C_cat H ) /\ x e. dom dom J ) -> ( ( Id ` C ) ` x ) = ( ( Id ` D ) ` x ) )
32	31	eleq1d 2533	. . . . . . 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 2828	. . . . . 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 6318	. . . . . . . 8 |- ( D |`cat J ) = ( ( C |`cat H ) |`cat J )
35	6	adantr 472	. . . . . . . . 9 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> C e. Cat )
36		dmexg 6743	. . . . . . . . . . 11 |- ( H e. (Subcat ` C ) -> dom H e. _V )
37		dmexg 6743	. . . . . . . . . . 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 472	. . . . . . . . 9 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> dom dom H e. _V )
40	35, 24, 28, 39, 29	rescabs 15816	. . . . . . . 8 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> ( ( C |`cat H ) |`cat J ) = ( C |`cat J ) )
41	34, 40	syl5req 2518	. . . . . . 7 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> ( C |`cat J ) = ( D |`cat J ) )
42	41	eleq1d 2533	. . . . . 6 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> ( ( C |`cat J ) e. Cat <-> ( D |`cat J ) e. Cat ) )
43	22, 33, 42	3anbi123d 1365	. . . . 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 2471	. . . . . 6 |- ( C |`cat J ) = ( C |`cat J )
45	17, 26, 44, 35, 28	issubc3 15832	. . . . 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 2471	. . . . . 6 |- ( Id ` D ) = ( Id ` D )
47		eqid 2471	. . . . . 6 |- ( D |`cat J ) = ( D |`cat J )
48	4, 7	subccat 15831	. . . . . . 7 |- ( H e. (Subcat ` C ) -> D e. Cat )
49	48	adantr 472	. . . . . 6 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> D e. Cat )
50	2, 46, 47, 49, 28	issubc3 15832	. . . . 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 294	. . . 4 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> ( J e. (Subcat ` D ) <-> J e. (Subcat ` C ) ) )
52	51	pm5.32da 653	. . 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 261	. 2 |- ( H e. (Subcat ` C ) -> ( J e. (Subcat ` D ) <-> ( J C_cat H /\ J e. (Subcat ` C ) ) ) )
54		ancom 457	. 2 |- ( ( J C_cat H /\ J e. (Subcat ` C ) ) <-> ( J e. (Subcat ` C ) /\ J C_cat H ) )
55	53, 54	syl6bb 269	1 |- ( H e. (Subcat ` C ) -> ( J e. (Subcat ` D ) <-> ( J e. (Subcat ` C ) /\ J C_cat H ) ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   A.wral 2756   _Vcvv 3031   class class class wbr 4395   X. cxp 4837   dom cdm 4839   Fn wfn 5584   ` cfv 5589  (class class class)co 6308   Basecbs 15199   Catccat 15648   Idccid 15649   Hom <sub>f</sub> chomf 15650   C_<sub>cat</sub> cssc 15790   |`_cat cresc 15791  Subcatcsubc 15792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  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-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-hom 15292  df-cco 15293  df-cat 15652  df-cid 15653  df-homf 15654  df-ssc 15793  df-resc 15794  df-subc 15795

This theorem is referenced by:  fldhmsubc  40594  fldhmsubcALTV  40613