Theorem subsubc 17935

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 2766	. . . . . 6 ⊢ (Homf ‘𝐷) = (Homf ‘𝐷)
3	1, 2	subcssc 17922	. . . . 5 ⊢ (𝐽 ∈ (Subcat‘𝐷) → 𝐽 ⊆cat (Homf ‘𝐷))
4		subsubc.d	. . . . . . 7 ⊢ 𝐷 = (𝐶 ↾cat 𝐻)
5		eqid 2766	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶)
6		subcrcl 17898	. . . . . . 7 ⊢ (𝐻 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat)
7		id 23	. . . . . . . 8 ⊢ (𝐻 ∈ (Subcat‘𝐶) → 𝐻 ∈ (Subcat‘𝐶))
8		eqidd 2767	. . . . . . . 8 ⊢ (𝐻 ∈ (Subcat‘𝐶) → dom dom 𝐻 = dom dom 𝐻)
9	7, 8	subcfn 17923	. . . . . . 7 ⊢ (𝐻 ∈ (Subcat‘𝐶) → 𝐻 Fn (dom dom 𝐻 × dom dom 𝐻))
10	7, 9, 5	subcss1 17924	. . . . . . 7 ⊢ (𝐻 ∈ (Subcat‘𝐶) → dom dom 𝐻 ⊆ (Base‘𝐶))
11	4, 5, 6, 9, 10	reschomf 17913	. . . . . 6 ⊢ (𝐻 ∈ (Subcat‘𝐶) → 𝐻 = (Homf ‘𝐷))
12	11	breq2d 5126	. . . . 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 572	. . 3 ⊢ (𝐻 ∈ (Subcat‘𝐶) → (𝐽 ∈ (Subcat‘𝐷) ↔ (𝐽 ⊆cat 𝐻 ∧ 𝐽 ∈ (Subcat‘𝐷))))
15		simpr 490	. . . . . . . 8 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → 𝐽 ⊆cat 𝐻)
16		simpl 488	. . . . . . . . 9 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → 𝐻 ∈ (Subcat‘𝐶))
17		eqid 2766	. . . . . . . . 9 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
18	16, 17	subcssc 17922	. . . . . . . 8 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → 𝐻 ⊆cat (Homf ‘𝐶))
19		ssctr 17907	. . . . . . . 8 ⊢ ((𝐽 ⊆cat 𝐻 ∧ 𝐻 ⊆cat (Homf ‘𝐶)) → 𝐽 ⊆cat (Homf ‘𝐶))
20	15, 18, 19	syl2anc 596	. . . . . . 7 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → 𝐽 ⊆cat (Homf ‘𝐶))
21	12	biimpa 482	. . . . . . 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 486	. . . . . . . . 9 ⊢ (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → 𝐻 ∈ (Subcat‘𝐶))
24	9	adantr 486	. . . . . . . . . 10 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → 𝐻 Fn (dom dom 𝐻 × dom dom 𝐻))
25	24	adantr 486	. . . . . . . . 9 ⊢ (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → 𝐻 Fn (dom dom 𝐻 × dom dom 𝐻))
26		eqid 2766	. . . . . . . . 9 ⊢ (Id‘𝐶) = (Id‘𝐶)
27		eqidd 2767	. . . . . . . . . . . 12 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → dom dom 𝐽 = dom dom 𝐽)
28	15, 27	sscfn1 17899	. . . . . . . . . . 11 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → 𝐽 Fn (dom dom 𝐽 × dom dom 𝐽))
29	28, 24, 15	ssc1 17903	. . . . . . . . . 10 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → dom dom 𝐽 ⊆ dom dom 𝐻)
30	29	sselda 3940	. . . . . . . . 9 ⊢ (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → 𝑥 ∈ dom dom 𝐻)
31	4, 23, 25, 26, 30	subcid 17929	. . . . . . . 8 ⊢ (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → ((Id‘𝐶)‘𝑥) = ((Id‘𝐷)‘𝑥))
32	31	eleq1d 2851	. . . . . . 7 ⊢ (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → (((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ↔ ((Id‘𝐷)‘𝑥) ∈ (𝑥𝐽𝑥)))
33	32	ralbidva 3189	. . . . . 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 7433	. . . . . . . 8 ⊢ (𝐷 ↾cat 𝐽) = ((𝐶 ↾cat 𝐻) ↾cat 𝐽)
35	6	adantr 486	. . . . . . . . 9 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → 𝐶 ∈ Cat)
36		dmexg 7907	. . . . . . . . . . 11 ⊢ (𝐻 ∈ (Subcat‘𝐶) → dom 𝐻 ∈ V)
37	36	dmexd 7909	. . . . . . . . . 10 ⊢ (𝐻 ∈ (Subcat‘𝐶) → dom dom 𝐻 ∈ V)
38	37	adantr 486	. . . . . . . . 9 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → dom dom 𝐻 ∈ V)
39	35, 24, 28, 38, 29	rescabs 17915	. . . . . . . 8 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → ((𝐶 ↾cat 𝐻) ↾cat 𝐽) = (𝐶 ↾cat 𝐽))
40	34, 39	eqtr2id 2814	. . . . . . 7 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → (𝐶 ↾cat 𝐽) = (𝐷 ↾cat 𝐽))
41	40	eleq1d 2851	. . . . . 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 2766	. . . . . 6 ⊢ (𝐶 ↾cat 𝐽) = (𝐶 ↾cat 𝐽)
44	17, 26, 43, 35, 28	issubc3 17931	. . . . 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 2766	. . . . . 6 ⊢ (Id‘𝐷) = (Id‘𝐷)
46		eqid 2766	. . . . . 6 ⊢ (𝐷 ↾cat 𝐽) = (𝐷 ↾cat 𝐽)
47	4, 7	subccat 17930	. . . . . . 7 ⊢ (𝐻 ∈ (Subcat‘𝐶) → 𝐷 ∈ Cat)
48	47	adantr 486	. . . . . 6 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → 𝐷 ∈ Cat)
49	2, 45, 46, 48, 28	issubc3 17931	. . . . 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 590	. . 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 469	1 ⊢ (𝐻 ∈ (Subcat‘𝐶) → (𝐽 ∈ (Subcat‘𝐷) ↔ (𝐽 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3082  Vcvv 3458   class class class wbr 5114   × cxp 5664  dom cdm 5666   Fn wfn 6538  ‘cfv 6543  (class class class)co 7423  Basecbs 17294  Catccat 17745  Idccid 17746  Hom_f chomf 17747   ⊆_cat cssc 17889   ↾_cat cresc 17890  Subcatcsubc 17891

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745  ax-cnex 11174  ax-resscn 11175  ax-1cn 11176  ax-icn 11177  ax-addcl 11178  ax-addrcl 11179  ax-mulcl 11180  ax-mulrcl 11181  ax-mulcom 11182  ax-addass 11183  ax-mulass 11184  ax-distr 11185  ax-i2m1 11186  ax-1ne0 11187  ax-1rid 11188  ax-rnegex 11189  ax-rrecex 11190  ax-cnre 11191  ax-pre-lttri 11192  ax-pre-lttrn 11193  ax-pre-ltadd 11194  ax-pre-mulgt0 11195

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-1st 7995  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-er 8703  df-pm 8836  df-ixp 8905  df-en 8953  df-dom 8954  df-sdom 8955  df-pnf 11263  df-mnf 11264  df-xr 11265  df-ltxr 11266  df-le 11267  df-sub 11461  df-neg 11462  df-nn 12252  df-2 12321  df-3 12322  df-4 12323  df-5 12324  df-6 12325  df-7 12326  df-8 12327  df-9 12328  df-n0 12523  df-z 12610  df-dec 12730  df-sets 17249  df-slot 17267  df-ndx 17279  df-base 17295  df-ress 17316  df-hom 17359  df-cco 17360  df-cat 17749  df-cid 17750  df-homf 17751  df-ssc 17892  df-resc 17893  df-subc 17894

This theorem is used by:  fldhmsubc  20925  fldhmsubcALTV  49139