Theorem subsubc 17911

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 23	. . . . . 6 ⊢ (𝐽 ∈ (Subcat‘𝐷) → 𝐽 ∈ (Subcat‘𝐷))
2		eqid 2763	. . . . . 6 ⊢ (Homf ‘𝐷) = (Homf ‘𝐷)
3	1, 2	subcssc 17898	. . . . 5 ⊢ (𝐽 ∈ (Subcat‘𝐷) → 𝐽 ⊆cat (Homf ‘𝐷))
4		subsubc.d	. . . . . . 7 ⊢ 𝐷 = (𝐶 ↾cat 𝐻)
5		eqid 2763	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶)
6		subcrcl 17874	. . . . . . 7 ⊢ (𝐻 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat)
7		id 23	. . . . . . . 8 ⊢ (𝐻 ∈ (Subcat‘𝐶) → 𝐻 ∈ (Subcat‘𝐶))
8		eqidd 2764	. . . . . . . 8 ⊢ (𝐻 ∈ (Subcat‘𝐶) → dom dom 𝐻 = dom dom 𝐻)
9	7, 8	subcfn 17899	. . . . . . 7 ⊢ (𝐻 ∈ (Subcat‘𝐶) → 𝐻 Fn (dom dom 𝐻 × dom dom 𝐻))
10	7, 9, 5	subcss1 17900	. . . . . . 7 ⊢ (𝐻 ∈ (Subcat‘𝐶) → dom dom 𝐻 ⊆ (Base‘𝐶))
11	4, 5, 6, 9, 10	reschomf 17889	. . . . . 6 ⊢ (𝐻 ∈ (Subcat‘𝐶) → 𝐻 = (Homf ‘𝐷))
12	11	breq2d 5122	. . . . 5 ⊢ (𝐻 ∈ (Subcat‘𝐶) → (𝐽 ⊆cat 𝐻 ↔ 𝐽 ⊆cat (Homf ‘𝐷)))
13	3, 12	imbitrrid 249	. . . 4 ⊢ (𝐻 ∈ (Subcat‘𝐶) → (𝐽 ∈ (Subcat‘𝐷) → 𝐽 ⊆cat 𝐻))
14	13	pm4.71rd 571	. . 3 ⊢ (𝐻 ∈ (Subcat‘𝐶) → (𝐽 ∈ (Subcat‘𝐷) ↔ (𝐽 ⊆cat 𝐻 ∧ 𝐽 ∈ (Subcat‘𝐷))))
15		simpr 489	. . . . . . . 8 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → 𝐽 ⊆cat 𝐻)
16		simpl 487	. . . . . . . . 9 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → 𝐻 ∈ (Subcat‘𝐶))
17		eqid 2763	. . . . . . . . 9 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
18	16, 17	subcssc 17898	. . . . . . . 8 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → 𝐻 ⊆cat (Homf ‘𝐶))
19		ssctr 17883	. . . . . . . 8 ⊢ ((𝐽 ⊆cat 𝐻 ∧ 𝐻 ⊆cat (Homf ‘𝐶)) → 𝐽 ⊆cat (Homf ‘𝐶))
20	15, 18, 19	syl2anc 595	. . . . . . 7 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → 𝐽 ⊆cat (Homf ‘𝐶))
21	12	biimpa 481	. . . . . . 7 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → 𝐽 ⊆cat (Homf ‘𝐷))
22	20, 21	2thd 268	. . . . . 6 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → (𝐽 ⊆cat (Homf ‘𝐶) ↔ 𝐽 ⊆cat (Homf ‘𝐷)))
23	16	adantr 485	. . . . . . . . 9 ⊢ (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → 𝐻 ∈ (Subcat‘𝐶))
24	9	adantr 485	. . . . . . . . . 10 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → 𝐻 Fn (dom dom 𝐻 × dom dom 𝐻))
25	24	adantr 485	. . . . . . . . 9 ⊢ (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → 𝐻 Fn (dom dom 𝐻 × dom dom 𝐻))
26		eqid 2763	. . . . . . . . 9 ⊢ (Id‘𝐶) = (Id‘𝐶)
27		eqidd 2764	. . . . . . . . . . . 12 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → dom dom 𝐽 = dom dom 𝐽)
28	15, 27	sscfn1 17875	. . . . . . . . . . 11 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → 𝐽 Fn (dom dom 𝐽 × dom dom 𝐽))
29	28, 24, 15	ssc1 17879	. . . . . . . . . 10 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → dom dom 𝐽 ⊆ dom dom 𝐻)
30	29	sselda 3938	. . . . . . . . 9 ⊢ (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → 𝑥 ∈ dom dom 𝐻)
31	4, 23, 25, 26, 30	subcid 17905	. . . . . . . 8 ⊢ (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → ((Id‘𝐶)‘𝑥) = ((Id‘𝐷)‘𝑥))
32	31	eleq1d 2848	. . . . . . 7 ⊢ (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → (((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ↔ ((Id‘𝐷)‘𝑥) ∈ (𝑥𝐽𝑥)))
33	32	ralbidva 3186	. . . . . 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 7422	. . . . . . . 8 ⊢ (𝐷 ↾cat 𝐽) = ((𝐶 ↾cat 𝐻) ↾cat 𝐽)
35	6	adantr 485	. . . . . . . . 9 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → 𝐶 ∈ Cat)
36		dmexg 7899	. . . . . . . . . . 11 ⊢ (𝐻 ∈ (Subcat‘𝐶) → dom 𝐻 ∈ V)
37	36	dmexd 7901	. . . . . . . . . 10 ⊢ (𝐻 ∈ (Subcat‘𝐶) → dom dom 𝐻 ∈ V)
38	37	adantr 485	. . . . . . . . 9 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → dom dom 𝐻 ∈ V)
39	35, 24, 28, 38, 29	rescabs 17891	. . . . . . . 8 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → ((𝐶 ↾cat 𝐻) ↾cat 𝐽) = (𝐶 ↾cat 𝐽))
40	34, 39	eqtr2id 2811	. . . . . . 7 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → (𝐶 ↾cat 𝐽) = (𝐷 ↾cat 𝐽))
41	40	eleq1d 2848	. . . . . 6 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → ((𝐶 ↾cat 𝐽) ∈ Cat ↔ (𝐷 ↾cat 𝐽) ∈ Cat))
42	22, 33, 41	3anbi123d 1464	. . . . 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 2763	. . . . . 6 ⊢ (𝐶 ↾cat 𝐽) = (𝐶 ↾cat 𝐽)
44	17, 26, 43, 35, 28	issubc3 17907	. . . . 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 2763	. . . . . 6 ⊢ (Id‘𝐷) = (Id‘𝐷)
46		eqid 2763	. . . . . 6 ⊢ (𝐷 ↾cat 𝐽) = (𝐷 ↾cat 𝐽)
47	4, 7	subccat 17906	. . . . . . 7 ⊢ (𝐻 ∈ (Subcat‘𝐶) → 𝐷 ∈ Cat)
48	47	adantr 485	. . . . . 6 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → 𝐷 ∈ Cat)
49	2, 45, 46, 48, 28	issubc3 17907	. . . . 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 315	. . . 4 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → (𝐽 ∈ (Subcat‘𝐷) ↔ 𝐽 ∈ (Subcat‘𝐶)))
51	50	pm5.32da 589	. . 3 ⊢ (𝐻 ∈ (Subcat‘𝐶) → ((𝐽 ⊆cat 𝐻 ∧ 𝐽 ∈ (Subcat‘𝐷)) ↔ (𝐽 ⊆cat 𝐻 ∧ 𝐽 ∈ (Subcat‘𝐶))))
52	14, 51	bitrd 282	. 2 ⊢ (𝐻 ∈ (Subcat‘𝐶) → (𝐽 ∈ (Subcat‘𝐷) ↔ (𝐽 ⊆cat 𝐻 ∧ 𝐽 ∈ (Subcat‘𝐶))))
53	52	biancomd 468	1 ⊢ (𝐻 ∈ (Subcat‘𝐶) → (𝐽 ∈ (Subcat‘𝐷) ↔ (𝐽 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079  Vcvv 3455   class class class wbr 5110   × cxp 5661  dom cdm 5663   Fn wfn 6533  ‘cfv 6538  (class class class)co 7412  Basecbs 17270  Catccat 17721  Idccid 17722  Hom_f chomf 17723   ⊆_cat cssc 17865   ↾_cat cresc 17866  Subcatcsubc 17867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-er 8695  df-pm 8828  df-ixp 8897  df-en 8945  df-dom 8946  df-sdom 8947  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-nn 12235  df-2 12304  df-3 12305  df-4 12306  df-5 12307  df-6 12308  df-7 12309  df-8 12310  df-9 12311  df-n0 12506  df-z 12593  df-dec 12713  df-sets 17225  df-slot 17243  df-ndx 17255  df-base 17271  df-ress 17292  df-hom 17335  df-cco 17336  df-cat 17725  df-cid 17726  df-homf 17727  df-ssc 17868  df-resc 17869  df-subc 17870

This theorem is referenced by:  fldhmsubc  20869  fldhmsubcALTV  49075