Theorem subsubc 17818

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 2740	. . . . . 6 ⊢ (Homf ‘𝐷) = (Homf ‘𝐷)
3	1, 2	subcssc 17805	. . . . 5 ⊢ (𝐽 ∈ (Subcat‘𝐷) → 𝐽 ⊆cat (Homf ‘𝐷))
4		subsubc.d	. . . . . . 7 ⊢ 𝐷 = (𝐶 ↾cat 𝐻)
5		eqid 2740	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶)
6		subcrcl 17781	. . . . . . 7 ⊢ (𝐻 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat)
7		id 22	. . . . . . . 8 ⊢ (𝐻 ∈ (Subcat‘𝐶) → 𝐻 ∈ (Subcat‘𝐶))
8		eqidd 2741	. . . . . . . 8 ⊢ (𝐻 ∈ (Subcat‘𝐶) → dom dom 𝐻 = dom dom 𝐻)
9	7, 8	subcfn 17806	. . . . . . 7 ⊢ (𝐻 ∈ (Subcat‘𝐶) → 𝐻 Fn (dom dom 𝐻 × dom dom 𝐻))
10	7, 9, 5	subcss1 17807	. . . . . . 7 ⊢ (𝐻 ∈ (Subcat‘𝐶) → dom dom 𝐻 ⊆ (Base‘𝐶))
11	4, 5, 6, 9, 10	reschomf 17796	. . . . . 6 ⊢ (𝐻 ∈ (Subcat‘𝐶) → 𝐻 = (Homf ‘𝐷))
12	11	breq2d 5091	. . . . 5 ⊢ (𝐻 ∈ (Subcat‘𝐶) → (𝐽 ⊆cat 𝐻 ↔ 𝐽 ⊆cat (Homf ‘𝐷)))
13	3, 12	imbitrrid 247	. . . 4 ⊢ (𝐻 ∈ (Subcat‘𝐶) → (𝐽 ∈ (Subcat‘𝐷) → 𝐽 ⊆cat 𝐻))
14	13	pm4.71rd 567	. . 3 ⊢ (𝐻 ∈ (Subcat‘𝐶) → (𝐽 ∈ (Subcat‘𝐷) ↔ (𝐽 ⊆cat 𝐻 ∧ 𝐽 ∈ (Subcat‘𝐷))))
15		simpr 485	. . . . . . . 8 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → 𝐽 ⊆cat 𝐻)
16		simpl 483	. . . . . . . . 9 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → 𝐻 ∈ (Subcat‘𝐶))
17		eqid 2740	. . . . . . . . 9 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
18	16, 17	subcssc 17805	. . . . . . . 8 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → 𝐻 ⊆cat (Homf ‘𝐶))
19		ssctr 17790	. . . . . . . 8 ⊢ ((𝐽 ⊆cat 𝐻 ∧ 𝐻 ⊆cat (Homf ‘𝐶)) → 𝐽 ⊆cat (Homf ‘𝐶))
20	15, 18, 19	syl2anc 590	. . . . . . 7 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → 𝐽 ⊆cat (Homf ‘𝐶))
21	12	biimpa 477	. . . . . . 7 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → 𝐽 ⊆cat (Homf ‘𝐷))
22	20, 21	2thd 266	. . . . . 6 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → (𝐽 ⊆cat (Homf ‘𝐶) ↔ 𝐽 ⊆cat (Homf ‘𝐷)))
23	16	adantr 481	. . . . . . . . 9 ⊢ (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → 𝐻 ∈ (Subcat‘𝐶))
24	9	adantr 481	. . . . . . . . . 10 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → 𝐻 Fn (dom dom 𝐻 × dom dom 𝐻))
25	24	adantr 481	. . . . . . . . 9 ⊢ (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → 𝐻 Fn (dom dom 𝐻 × dom dom 𝐻))
26		eqid 2740	. . . . . . . . 9 ⊢ (Id‘𝐶) = (Id‘𝐶)
27		eqidd 2741	. . . . . . . . . . . 12 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → dom dom 𝐽 = dom dom 𝐽)
28	15, 27	sscfn1 17782	. . . . . . . . . . 11 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → 𝐽 Fn (dom dom 𝐽 × dom dom 𝐽))
29	28, 24, 15	ssc1 17786	. . . . . . . . . 10 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → dom dom 𝐽 ⊆ dom dom 𝐻)
30	29	sselda 3922	. . . . . . . . 9 ⊢ (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → 𝑥 ∈ dom dom 𝐻)
31	4, 23, 25, 26, 30	subcid 17812	. . . . . . . 8 ⊢ (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → ((Id‘𝐶)‘𝑥) = ((Id‘𝐷)‘𝑥))
32	31	eleq1d 2825	. . . . . . 7 ⊢ (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → (((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ↔ ((Id‘𝐷)‘𝑥) ∈ (𝑥𝐽𝑥)))
33	32	ralbidva 3161	. . . . . 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 7373	. . . . . . . 8 ⊢ (𝐷 ↾cat 𝐽) = ((𝐶 ↾cat 𝐻) ↾cat 𝐽)
35	6	adantr 481	. . . . . . . . 9 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → 𝐶 ∈ Cat)
36		dmexg 7848	. . . . . . . . . . 11 ⊢ (𝐻 ∈ (Subcat‘𝐶) → dom 𝐻 ∈ V)
37	36	dmexd 7850	. . . . . . . . . 10 ⊢ (𝐻 ∈ (Subcat‘𝐶) → dom dom 𝐻 ∈ V)
38	37	adantr 481	. . . . . . . . 9 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → dom dom 𝐻 ∈ V)
39	35, 24, 28, 38, 29	rescabs 17798	. . . . . . . 8 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → ((𝐶 ↾cat 𝐻) ↾cat 𝐽) = (𝐶 ↾cat 𝐽))
40	34, 39	eqtr2id 2788	. . . . . . 7 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → (𝐶 ↾cat 𝐽) = (𝐷 ↾cat 𝐽))
41	40	eleq1d 2825	. . . . . 6 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → ((𝐶 ↾cat 𝐽) ∈ Cat ↔ (𝐷 ↾cat 𝐽) ∈ Cat))
42	22, 33, 41	3anbi123d 1444	. . . . 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 2740	. . . . . 6 ⊢ (𝐶 ↾cat 𝐽) = (𝐶 ↾cat 𝐽)
44	17, 26, 43, 35, 28	issubc3 17814	. . . . 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 2740	. . . . . 6 ⊢ (Id‘𝐷) = (Id‘𝐷)
46		eqid 2740	. . . . . 6 ⊢ (𝐷 ↾cat 𝐽) = (𝐷 ↾cat 𝐽)
47	4, 7	subccat 17813	. . . . . . 7 ⊢ (𝐻 ∈ (Subcat‘𝐶) → 𝐷 ∈ Cat)
48	47	adantr 481	. . . . . 6 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → 𝐷 ∈ Cat)
49	2, 45, 46, 48, 28	issubc3 17814	. . . . 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 313	. . . 4 ⊢ ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻) → (𝐽 ∈ (Subcat‘𝐷) ↔ 𝐽 ∈ (Subcat‘𝐶)))
51	50	pm5.32da 584	. . 3 ⊢ (𝐻 ∈ (Subcat‘𝐶) → ((𝐽 ⊆cat 𝐻 ∧ 𝐽 ∈ (Subcat‘𝐷)) ↔ (𝐽 ⊆cat 𝐻 ∧ 𝐽 ∈ (Subcat‘𝐶))))
52	14, 51	bitrd 280	. 2 ⊢ (𝐻 ∈ (Subcat‘𝐶) → (𝐽 ∈ (Subcat‘𝐷) ↔ (𝐽 ⊆cat 𝐻 ∧ 𝐽 ∈ (Subcat‘𝐶))))
53	52	biancomd 464	1 ⊢ (𝐻 ∈ (Subcat‘𝐶) → (𝐽 ∈ (Subcat‘𝐷) ↔ (𝐽 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3054  Vcvv 3432   class class class wbr 5079   × cxp 5623  dom cdm 5625   Fn wfn 6487  ‘cfv 6492  (class class class)co 7363  Basecbs 17177  Catccat 17628  Idccid 17629  Hom_f chomf 17630   ⊆_cat cssc 17772   ↾_cat cresc 17773  Subcatcsubc 17774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-er 8640  df-pm 8773  df-ixp 8843  df-en 8891  df-dom 8892  df-sdom 8893  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-nn 12173  df-2 12242  df-3 12243  df-4 12244  df-5 12245  df-6 12246  df-7 12247  df-8 12248  df-9 12249  df-n0 12436  df-z 12523  df-dec 12643  df-sets 17132  df-slot 17150  df-ndx 17162  df-base 17178  df-ress 17199  df-hom 17242  df-cco 17243  df-cat 17632  df-cid 17633  df-homf 17634  df-ssc 17775  df-resc 17776  df-subc 17777

This theorem is referenced by:  fldhmsubc  20764  fldhmsubcALTV  48831