Theorem subsubc 17549

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 2739	. . . . . 6 ⊢ (Homf ‘𝐷) = (Homf ‘𝐷)
3	1, 2	subcssc 17536	. . . . 5 ⊢ (𝐽 ∈ (Subcat‘𝐷) → 𝐽 ⊆cat (Homf ‘𝐷))
4		subsubc.d	. . . . . . 7 ⊢ 𝐷 = (𝐶 ↾cat 𝐻)
5		eqid 2739	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶)
6		subcrcl 17509	. . . . . . 7 ⊢ (𝐻 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat)
7		id 22	. . . . . . . 8 ⊢ (𝐻 ∈ (Subcat‘𝐶) → 𝐻 ∈ (Subcat‘𝐶))
8		eqidd 2740	. . . . . . . 8 ⊢ (𝐻 ∈ (Subcat‘𝐶) → dom dom 𝐻 = dom dom 𝐻)
9	7, 8	subcfn 17537	. . . . . . 7 ⊢ (𝐻 ∈ (Subcat‘𝐶) → 𝐻 Fn (dom dom 𝐻 × dom dom 𝐻))
10	7, 9, 5	subcss1 17538	. . . . . . 7 ⊢ (𝐻 ∈ (Subcat‘𝐶) → dom dom 𝐻 ⊆ (Base‘𝐶))
11	4, 5, 6, 9, 10	reschomf 17525	. . . . . 6 ⊢ (𝐻 ∈ (Subcat‘𝐶) → 𝐻 = (Homf ‘𝐷))
12	11	breq2d 5090	. . . . 5 ⊢ (𝐻 ∈ (Subcat‘𝐶) → (𝐽 ⊆cat 𝐻 ↔ 𝐽 ⊆cat (Homf ‘𝐷)))
13	3, 12	syl5ibr 245	. . . 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 2739	. . . . . . . . 9 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
18	16, 17	subcssc 17536	. . . . . . . 8 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → 𝐻 ⊆cat (Homf ‘𝐶))
19		ssctr 17518	. . . . . . . 8 ⊢ ((𝐽 ⊆cat 𝐻 ∧ 𝐻 ⊆cat (Homf ‘𝐶)) → 𝐽 ⊆cat (Homf ‘𝐶))
20	15, 18, 19	syl2anc 583	. . . . . . 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 264	. . . . . 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 2739	. . . . . . . . 9 ⊢ (Id‘𝐶) = (Id‘𝐶)
27		eqidd 2740	. . . . . . . . . . . 12 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → dom dom 𝐽 = dom dom 𝐽)
28	15, 27	sscfn1 17510	. . . . . . . . . . 11 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → 𝐽 Fn (dom dom 𝐽 × dom dom 𝐽))
29	28, 24, 15	ssc1 17514	. . . . . . . . . 10 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → dom dom 𝐽 ⊆ dom dom 𝐻)
30	29	sselda 3925	. . . . . . . . 9 ⊢ (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → 𝑥 ∈ dom dom 𝐻)
31	4, 23, 25, 26, 30	subcid 17543	. . . . . . . 8 ⊢ (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → ((Id‘𝐶)‘𝑥) = ((Id‘𝐷)‘𝑥))
32	31	eleq1d 2824	. . . . . . 7 ⊢ (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → (((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ↔ ((Id‘𝐷)‘𝑥) ∈ (𝑥𝐽𝑥)))
33	32	ralbidva 3121	. . . . . 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 7278	. . . . . . . 8 ⊢ (𝐷 ↾cat 𝐽) = ((𝐶 ↾cat 𝐻) ↾cat 𝐽)
35	6	adantr 480	. . . . . . . . 9 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → 𝐶 ∈ Cat)
36		dmexg 7737	. . . . . . . . . . 11 ⊢ (𝐻 ∈ (Subcat‘𝐶) → dom 𝐻 ∈ V)
37	36	dmexd 7739	. . . . . . . . . 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 17528	. . . . . . . 8 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → ((𝐶 ↾cat 𝐻) ↾cat 𝐽) = (𝐶 ↾cat 𝐽))
40	34, 39	eqtr2id 2792	. . . . . . 7 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → (𝐶 ↾cat 𝐽) = (𝐷 ↾cat 𝐽))
41	40	eleq1d 2824	. . . . . 6 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → ((𝐶 ↾cat 𝐽) ∈ Cat ↔ (𝐷 ↾cat 𝐽) ∈ Cat))
42	22, 33, 41	3anbi123d 1434	. . . . 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 2739	. . . . . 6 ⊢ (𝐶 ↾cat 𝐽) = (𝐶 ↾cat 𝐽)
44	17, 26, 43, 35, 28	issubc3 17545	. . . . 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 2739	. . . . . 6 ⊢ (Id‘𝐷) = (Id‘𝐷)
46		eqid 2739	. . . . . 6 ⊢ (𝐷 ↾cat 𝐽) = (𝐷 ↾cat 𝐽)
47	4, 7	subccat 17544	. . . . . . 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 17545	. . . . 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 311	. . . 4 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → (𝐽 ∈ (Subcat‘𝐷) ↔ 𝐽 ∈ (Subcat‘𝐶)))
51	50	pm5.32da 578	. . 3 ⊢ (𝐻 ∈ (Subcat‘𝐶) → ((𝐽 ⊆cat 𝐻 ∧ 𝐽 ∈ (Subcat‘𝐷)) ↔ (𝐽 ⊆cat 𝐻 ∧ 𝐽 ∈ (Subcat‘𝐶))))
52	14, 51	bitrd 278	. 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 205   ∧ wa 395   ∧ w3a 1085   = wceq 1541   ∈ wcel 2109  ∀wral 3065  Vcvv 3430   class class class wbr 5078   × cxp 5586  dom cdm 5588   Fn wfn 6425  ‘cfv 6430  (class class class)co 7268  Basecbs 16893  Catccat 17354  Idccid 17355  Hom_f chomf 17356   ⊆_cat cssc 17500   ↾_cat cresc 17501  Subcatcsubc 17502

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579  ax-cnex 10911  ax-resscn 10912  ax-1cn 10913  ax-icn 10914  ax-addcl 10915  ax-addrcl 10916  ax-mulcl 10917  ax-mulrcl 10918  ax-mulcom 10919  ax-addass 10920  ax-mulass 10921  ax-distr 10922  ax-i2m1 10923  ax-1ne0 10924  ax-1rid 10925  ax-rnegex 10926  ax-rrecex 10927  ax-cnre 10928  ax-pre-lttri 10929  ax-pre-lttrn 10930  ax-pre-ltadd 10931  ax-pre-mulgt0 10932

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3070  df-rex 3071  df-reu 3072  df-rmo 3073  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-pss 3910  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4845  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-tr 5196  df-id 5488  df-eprel 5494  df-po 5502  df-so 5503  df-fr 5543  df-we 5545  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-pred 6199  df-ord 6266  df-on 6267  df-lim 6268  df-suc 6269  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-riota 7225  df-ov 7271  df-oprab 7272  df-mpo 7273  df-om 7701  df-1st 7817  df-2nd 7818  df-frecs 8081  df-wrecs 8112  df-recs 8186  df-rdg 8225  df-er 8472  df-pm 8592  df-ixp 8660  df-en 8708  df-dom 8709  df-sdom 8710  df-pnf 10995  df-mnf 10996  df-xr 10997  df-ltxr 10998  df-le 10999  df-sub 11190  df-neg 11191  df-nn 11957  df-2 12019  df-3 12020  df-4 12021  df-5 12022  df-6 12023  df-7 12024  df-8 12025  df-9 12026  df-n0 12217  df-z 12303  df-dec 12420  df-sets 16846  df-slot 16864  df-ndx 16876  df-base 16894  df-ress 16923  df-hom 16967  df-cco 16968  df-cat 17358  df-cid 17359  df-homf 17360  df-ssc 17503  df-resc 17504  df-subc 17505

This theorem is referenced by:  fldhmsubc  45594  fldhmsubcALTV  45612