Theorem subsubc 17757

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 ∈ (Subcat‘𝐶) → (𝐽 ∈ (Subcat‘𝐷) ↔ (𝐽 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻)))

Proof of Theorem subsubc

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . . . . 6 ⊢ (𝐽 ∈ (Subcat‘𝐷) → 𝐽 ∈ (Subcat‘𝐷))
2		eqid 2731	. . . . . 6 ⊢ (Homf ‘𝐷) = (Homf ‘𝐷)
3	1, 2	subcssc 17744	. . . . 5 ⊢ (𝐽 ∈ (Subcat‘𝐷) → 𝐽 ⊆cat (Homf ‘𝐷))
4		subsubc.d	. . . . . . 7 ⊢ 𝐷 = (𝐶 ↾cat 𝐻)
5		eqid 2731	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶)
6		subcrcl 17720	. . . . . . 7 ⊢ (𝐻 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat)
7		id 22	. . . . . . . 8 ⊢ (𝐻 ∈ (Subcat‘𝐶) → 𝐻 ∈ (Subcat‘𝐶))
8		eqidd 2732	. . . . . . . 8 ⊢ (𝐻 ∈ (Subcat‘𝐶) → dom dom 𝐻 = dom dom 𝐻)
9	7, 8	subcfn 17745	. . . . . . 7 ⊢ (𝐻 ∈ (Subcat‘𝐶) → 𝐻 Fn (dom dom 𝐻 × dom dom 𝐻))
10	7, 9, 5	subcss1 17746	. . . . . . 7 ⊢ (𝐻 ∈ (Subcat‘𝐶) → dom dom 𝐻 ⊆ (Base‘𝐶))
11	4, 5, 6, 9, 10	reschomf 17735	. . . . . 6 ⊢ (𝐻 ∈ (Subcat‘𝐶) → 𝐻 = (Homf ‘𝐷))
12	11	breq2d 5103	. . . . 5 ⊢ (𝐻 ∈ (Subcat‘𝐶) → (𝐽 ⊆cat 𝐻 ↔ 𝐽 ⊆cat (Homf ‘𝐷)))
13	3, 12	imbitrrid 246	. . . 4 ⊢ (𝐻 ∈ (Subcat‘𝐶) → (𝐽 ∈ (Subcat‘𝐷) → 𝐽 ⊆cat 𝐻))
14	13	pm4.71rd 562	. . 3 ⊢ (𝐻 ∈ (Subcat‘𝐶) → (𝐽 ∈ (Subcat‘𝐷) ↔ (𝐽 ⊆cat 𝐻 ∧ 𝐽 ∈ (Subcat‘𝐷))))
15		simpr 484	. . . . . . . 8 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → 𝐽 ⊆cat 𝐻)
16		simpl 482	. . . . . . . . 9 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → 𝐻 ∈ (Subcat‘𝐶))
17		eqid 2731	. . . . . . . . 9 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
18	16, 17	subcssc 17744	. . . . . . . 8 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → 𝐻 ⊆cat (Homf ‘𝐶))
19		ssctr 17729	. . . . . . . 8 ⊢ ((𝐽 ⊆cat 𝐻 ∧ 𝐻 ⊆cat (Homf ‘𝐶)) → 𝐽 ⊆cat (Homf ‘𝐶))
20	15, 18, 19	syl2anc 584	. . . . . . 7 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → 𝐽 ⊆cat (Homf ‘𝐶))
21	12	biimpa 476	. . . . . . 7 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → 𝐽 ⊆cat (Homf ‘𝐷))
22	20, 21	2thd 265	. . . . . 6 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → (𝐽 ⊆cat (Homf ‘𝐶) ↔ 𝐽 ⊆cat (Homf ‘𝐷)))
23	16	adantr 480	. . . . . . . . 9 ⊢ (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → 𝐻 ∈ (Subcat‘𝐶))
24	9	adantr 480	. . . . . . . . . 10 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → 𝐻 Fn (dom dom 𝐻 × dom dom 𝐻))
25	24	adantr 480	. . . . . . . . 9 ⊢ (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → 𝐻 Fn (dom dom 𝐻 × dom dom 𝐻))
26		eqid 2731	. . . . . . . . 9 ⊢ (Id‘𝐶) = (Id‘𝐶)
27		eqidd 2732	. . . . . . . . . . . 12 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → dom dom 𝐽 = dom dom 𝐽)
28	15, 27	sscfn1 17721	. . . . . . . . . . 11 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → 𝐽 Fn (dom dom 𝐽 × dom dom 𝐽))
29	28, 24, 15	ssc1 17725	. . . . . . . . . 10 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → dom dom 𝐽 ⊆ dom dom 𝐻)
30	29	sselda 3934	. . . . . . . . 9 ⊢ (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → 𝑥 ∈ dom dom 𝐻)
31	4, 23, 25, 26, 30	subcid 17751	. . . . . . . 8 ⊢ (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → ((Id‘𝐶)‘𝑥) = ((Id‘𝐷)‘𝑥))
32	31	eleq1d 2816	. . . . . . 7 ⊢ (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → (((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ↔ ((Id‘𝐷)‘𝑥) ∈ (𝑥𝐽𝑥)))
33	32	ralbidva 3153	. . . . . 6 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → (∀𝑥 ∈ dom dom 𝐽((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ↔ ∀𝑥 ∈ dom dom 𝐽((Id‘𝐷)‘𝑥) ∈ (𝑥𝐽𝑥)))
34	4	oveq1i 7356	. . . . . . . 8 ⊢ (𝐷 ↾cat 𝐽) = ((𝐶 ↾cat 𝐻) ↾cat 𝐽)
35	6	adantr 480	. . . . . . . . 9 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → 𝐶 ∈ Cat)
36		dmexg 7831	. . . . . . . . . . 11 ⊢ (𝐻 ∈ (Subcat‘𝐶) → dom 𝐻 ∈ V)
37	36	dmexd 7833	. . . . . . . . . 10 ⊢ (𝐻 ∈ (Subcat‘𝐶) → dom dom 𝐻 ∈ V)
38	37	adantr 480	. . . . . . . . 9 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → dom dom 𝐻 ∈ V)
39	35, 24, 28, 38, 29	rescabs 17737	. . . . . . . 8 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → ((𝐶 ↾cat 𝐻) ↾cat 𝐽) = (𝐶 ↾cat 𝐽))
40	34, 39	eqtr2id 2779	. . . . . . 7 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → (𝐶 ↾cat 𝐽) = (𝐷 ↾cat 𝐽))
41	40	eleq1d 2816	. . . . . 6 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → ((𝐶 ↾cat 𝐽) ∈ Cat ↔ (𝐷 ↾cat 𝐽) ∈ Cat))
42	22, 33, 41	3anbi123d 1438	. . . . 5 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → ((𝐽 ⊆cat (Homf ‘𝐶) ∧ ∀𝑥 ∈ dom dom 𝐽((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ∧ (𝐶 ↾cat 𝐽) ∈ Cat) ↔ (𝐽 ⊆cat (Homf ‘𝐷) ∧ ∀𝑥 ∈ dom dom 𝐽((Id‘𝐷)‘𝑥) ∈ (𝑥𝐽𝑥) ∧ (𝐷 ↾cat 𝐽) ∈ Cat)))
43		eqid 2731	. . . . . 6 ⊢ (𝐶 ↾cat 𝐽) = (𝐶 ↾cat 𝐽)
44	17, 26, 43, 35, 28	issubc3 17753	. . . . 5 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽 ⊆cat (Homf ‘𝐶) ∧ ∀𝑥 ∈ dom dom 𝐽((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ∧ (𝐶 ↾cat 𝐽) ∈ Cat)))
45		eqid 2731	. . . . . 6 ⊢ (Id‘𝐷) = (Id‘𝐷)
46		eqid 2731	. . . . . 6 ⊢ (𝐷 ↾cat 𝐽) = (𝐷 ↾cat 𝐽)
47	4, 7	subccat 17752	. . . . . . 7 ⊢ (𝐻 ∈ (Subcat‘𝐶) → 𝐷 ∈ Cat)
48	47	adantr 480	. . . . . 6 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → 𝐷 ∈ Cat)
49	2, 45, 46, 48, 28	issubc3 17753	. . . . 5 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → (𝐽 ∈ (Subcat‘𝐷) ↔ (𝐽 ⊆cat (Homf ‘𝐷) ∧ ∀𝑥 ∈ dom dom 𝐽((Id‘𝐷)‘𝑥) ∈ (𝑥𝐽𝑥) ∧ (𝐷 ↾cat 𝐽) ∈ Cat)))
50	42, 44, 49	3bitr4rd 312	. . . 4 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → (𝐽 ∈ (Subcat‘𝐷) ↔ 𝐽 ∈ (Subcat‘𝐶)))
51	50	pm5.32da 579	. . 3 ⊢ (𝐻 ∈ (Subcat‘𝐶) → ((𝐽 ⊆cat 𝐻 ∧ 𝐽 ∈ (Subcat‘𝐷)) ↔ (𝐽 ⊆cat 𝐻 ∧ 𝐽 ∈ (Subcat‘𝐶))))
52	14, 51	bitrd 279	. 2 ⊢ (𝐻 ∈ (Subcat‘𝐶) → (𝐽 ∈ (Subcat‘𝐷) ↔ (𝐽 ⊆cat 𝐻 ∧ 𝐽 ∈ (Subcat‘𝐶))))
53	52	biancomd 463	1 ⊢ (𝐻 ∈ (Subcat‘𝐶) → (𝐽 ∈ (Subcat‘𝐷) ↔ (𝐽 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∀wral 3047  Vcvv 3436   class class class wbr 5091   × cxp 5614  dom cdm 5616   Fn wfn 6476  ‘cfv 6481  (class class class)co 7346  Basecbs 17117  Catccat 17567  Idccid 17568  Hom_f chomf 17569   ⊆_cat cssc 17711   ↾_cat cresc 17712  Subcatcsubc 17713

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-cnex 11059  ax-resscn 11060  ax-1cn 11061  ax-icn 11062  ax-addcl 11063  ax-addrcl 11064  ax-mulcl 11065  ax-mulrcl 11066  ax-mulcom 11067  ax-addass 11068  ax-mulass 11069  ax-distr 11070  ax-i2m1 11071  ax-1ne0 11072  ax-1rid 11073  ax-rnegex 11074  ax-rrecex 11075  ax-cnre 11076  ax-pre-lttri 11077  ax-pre-lttrn 11078  ax-pre-ltadd 11079  ax-pre-mulgt0 11080

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-er 8622  df-pm 8753  df-ixp 8822  df-en 8870  df-dom 8871  df-sdom 8872  df-pnf 11145  df-mnf 11146  df-xr 11147  df-ltxr 11148  df-le 11149  df-sub 11343  df-neg 11344  df-nn 12123  df-2 12185  df-3 12186  df-4 12187  df-5 12188  df-6 12189  df-7 12190  df-8 12191  df-9 12192  df-n0 12379  df-z 12466  df-dec 12586  df-sets 17072  df-slot 17090  df-ndx 17102  df-base 17118  df-ress 17139  df-hom 17182  df-cco 17183  df-cat 17571  df-cid 17572  df-homf 17573  df-ssc 17714  df-resc 17715  df-subc 17716

This theorem is referenced by:  fldhmsubc  20698  fldhmsubcALTV  48363