Theorem ressffth 17974

Description: The inclusion functor from a full subcategory is a full and faithful functor, see also remark 4.4(2) in [Adamek] p. 49. (Contributed by Mario Carneiro, 27-Jan-2017.)

Hypotheses

Ref	Expression
ressffth.d	⊢ 𝐷 = (𝐶 ↾s 𝑆)
ressffth.i	⊢ 𝐼 = (idfunc‘𝐷)

Assertion

Ref	Expression
ressffth	⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → 𝐼 ∈ ((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶)))

Proof of Theorem ressffth

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relfunc 17896	. . 3 ⊢ Rel (𝐷 Func 𝐷)
2		ressffth.d	. . . . 5 ⊢ 𝐷 = (𝐶 ↾s 𝑆)
3		resscat 17886	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (𝐶 ↾s 𝑆) ∈ Cat)
4	2, 3	eqeltrid 2867	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → 𝐷 ∈ Cat)
5		ressffth.i	. . . . 5 ⊢ 𝐼 = (idfunc‘𝐷)
6	5	idfucl 17915	. . . 4 ⊢ (𝐷 ∈ Cat → 𝐼 ∈ (𝐷 Func 𝐷))
7	4, 6	syl 17	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → 𝐼 ∈ (𝐷 Func 𝐷))
8		1st2nd 8021	. . 3 ⊢ ((Rel (𝐷 Func 𝐷) ∧ 𝐼 ∈ (𝐷 Func 𝐷)) → 𝐼 = ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩)
9	1, 7, 8	sylancr 596	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → 𝐼 = ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩)
10		eqidd 2764	. . . . . . . . 9 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (Homf ‘𝐷) = (Homf ‘𝐷))
11		eqidd 2764	. . . . . . . . 9 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (compf‘𝐷) = (compf‘𝐷))
12		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐶) = (Base‘𝐶)
13	12	ressinbas 17282	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ 𝑉 → (𝐶 ↾s 𝑆) = (𝐶 ↾s (𝑆 ∩ (Base‘𝐶))))
14	13	adantl 485	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (𝐶 ↾s 𝑆) = (𝐶 ↾s (𝑆 ∩ (Base‘𝐶))))
15	2, 14	eqtrid 2810	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → 𝐷 = (𝐶 ↾s (𝑆 ∩ (Base‘𝐶))))
16	15	fveq2d 6872	. . . . . . . . . 10 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (Homf ‘𝐷) = (Homf ‘(𝐶 ↾s (𝑆 ∩ (Base‘𝐶)))))
17		eqid 2763	. . . . . . . . . . . 12 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
18		simpl 486	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → 𝐶 ∈ Cat)
19		inss2 4190	. . . . . . . . . . . . 13 ⊢ (𝑆 ∩ (Base‘𝐶)) ⊆ (Base‘𝐶)
20	19	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (𝑆 ∩ (Base‘𝐶)) ⊆ (Base‘𝐶))
21		eqid 2763	. . . . . . . . . . . 12 ⊢ (𝐶 ↾s (𝑆 ∩ (Base‘𝐶))) = (𝐶 ↾s (𝑆 ∩ (Base‘𝐶)))
22		eqid 2763	. . . . . . . . . . . 12 ⊢ (𝐶 ↾cat ((Homf ‘𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶))))) = (𝐶 ↾cat ((Homf ‘𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))
23	12, 17, 18, 20, 21, 22	fullresc 17885	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → ((Homf ‘(𝐶 ↾s (𝑆 ∩ (Base‘𝐶)))) = (Homf ‘(𝐶 ↾cat ((Homf ‘𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))) ∧ (compf‘(𝐶 ↾s (𝑆 ∩ (Base‘𝐶)))) = (compf‘(𝐶 ↾cat ((Homf ‘𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶))))))))
24	23	simpld 498	. . . . . . . . . 10 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (Homf ‘(𝐶 ↾s (𝑆 ∩ (Base‘𝐶)))) = (Homf ‘(𝐶 ↾cat ((Homf ‘𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))))
25	16, 24	eqtrd 2798	. . . . . . . . 9 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (Homf ‘𝐷) = (Homf ‘(𝐶 ↾cat ((Homf ‘𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))))
26	15	fveq2d 6872	. . . . . . . . . 10 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (compf‘𝐷) = (compf‘(𝐶 ↾s (𝑆 ∩ (Base‘𝐶)))))
27	23	simprd 499	. . . . . . . . . 10 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (compf‘(𝐶 ↾s (𝑆 ∩ (Base‘𝐶)))) = (compf‘(𝐶 ↾cat ((Homf ‘𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))))
28	26, 27	eqtrd 2798	. . . . . . . . 9 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (compf‘𝐷) = (compf‘(𝐶 ↾cat ((Homf ‘𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))))
29	2	ovexi 7431	. . . . . . . . . 10 ⊢ 𝐷 ∈ V
30	29	a1i 11	. . . . . . . . 9 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → 𝐷 ∈ V)
31		ovexd 7432	. . . . . . . . 9 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (𝐶 ↾cat ((Homf ‘𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶))))) ∈ V)
32	10, 11, 25, 28, 30, 30, 30, 31	funcpropd 17936	. . . . . . . 8 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (𝐷 Func 𝐷) = (𝐷 Func (𝐶 ↾cat ((Homf ‘𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))))
33	12, 17, 18, 20	fullsubc 17884	. . . . . . . . 9 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → ((Homf ‘𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))) ∈ (Subcat‘𝐶))
34		funcres2 17932	. . . . . . . . 9 ⊢ (((Homf ‘𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))) ∈ (Subcat‘𝐶) → (𝐷 Func (𝐶 ↾cat ((Homf ‘𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))) ⊆ (𝐷 Func 𝐶))
35	33, 34	syl 17	. . . . . . . 8 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (𝐷 Func (𝐶 ↾cat ((Homf ‘𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))) ⊆ (𝐷 Func 𝐶))
36	32, 35	eqsstrd 3971	. . . . . . 7 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (𝐷 Func 𝐷) ⊆ (𝐷 Func 𝐶))
37	36, 7	sseldd 3938	. . . . . 6 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → 𝐼 ∈ (𝐷 Func 𝐶))
38	9, 37	eqeltrrd 2864	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩ ∈ (𝐷 Func 𝐶))
39		df-br 5102	. . . . 5 ⊢ ((1st ‘𝐼)(𝐷 Func 𝐶)(2nd ‘𝐼) ↔ ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩ ∈ (𝐷 Func 𝐶))
40	38, 39	sylibr 236	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (1st ‘𝐼)(𝐷 Func 𝐶)(2nd ‘𝐼))
41		f1oi 6846	. . . . . 6 ⊢ ( I ↾ (𝑥(Hom ‘𝐷)𝑦)):(𝑥(Hom ‘𝐷)𝑦)–1-1-onto→(𝑥(Hom ‘𝐷)𝑦)
42		eqid 2763	. . . . . . . 8 ⊢ (Base‘𝐷) = (Base‘𝐷)
43	4	adantr 484	. . . . . . . 8 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐷 ∈ Cat)
44		eqid 2763	. . . . . . . 8 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
45		simprl 780	. . . . . . . 8 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑥 ∈ (Base‘𝐷))
46		simprr 782	. . . . . . . 8 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑦 ∈ (Base‘𝐷))
47	5, 42, 43, 44, 45, 46	idfu2nd 17911	. . . . . . 7 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑥(2nd ‘𝐼)𝑦) = ( I ↾ (𝑥(Hom ‘𝐷)𝑦)))
48		eqidd 2764	. . . . . . 7 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
49		eqid 2763	. . . . . . . . . 10 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
50	2, 49	resshom 17448	. . . . . . . . 9 ⊢ (𝑆 ∈ 𝑉 → (Hom ‘𝐶) = (Hom ‘𝐷))
51	50	ad2antlr 737	. . . . . . . 8 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (Hom ‘𝐶) = (Hom ‘𝐷))
52	5, 42, 43, 45	idfu1 17914	. . . . . . . 8 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → ((1st ‘𝐼)‘𝑥) = 𝑥)
53	5, 42, 43, 46	idfu1 17914	. . . . . . . 8 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → ((1st ‘𝐼)‘𝑦) = 𝑦)
54	51, 52, 53	oveq123d 7418	. . . . . . 7 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (((1st ‘𝐼)‘𝑥)(Hom ‘𝐶)((1st ‘𝐼)‘𝑦)) = (𝑥(Hom ‘𝐷)𝑦))
55	47, 48, 54	f1oeq123d 6801	. . . . . 6 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → ((𝑥(2nd ‘𝐼)𝑦):(𝑥(Hom ‘𝐷)𝑦)–1-1-onto→(((1st ‘𝐼)‘𝑥)(Hom ‘𝐶)((1st ‘𝐼)‘𝑦)) ↔ ( I ↾ (𝑥(Hom ‘𝐷)𝑦)):(𝑥(Hom ‘𝐷)𝑦)–1-1-onto→(𝑥(Hom ‘𝐷)𝑦)))
56	41, 55	mpbiri 260	. . . . 5 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑥(2nd ‘𝐼)𝑦):(𝑥(Hom ‘𝐷)𝑦)–1-1-onto→(((1st ‘𝐼)‘𝑥)(Hom ‘𝐶)((1st ‘𝐼)‘𝑦)))
57	56	ralrimivva 3206	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → ∀𝑥 ∈ (Base‘𝐷)∀𝑦 ∈ (Base‘𝐷)(𝑥(2nd ‘𝐼)𝑦):(𝑥(Hom ‘𝐷)𝑦)–1-1-onto→(((1st ‘𝐼)‘𝑥)(Hom ‘𝐶)((1st ‘𝐼)‘𝑦)))
58	42, 44, 49	isffth2 17952	. . . 4 ⊢ ((1st ‘𝐼)((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶))(2nd ‘𝐼) ↔ ((1st ‘𝐼)(𝐷 Func 𝐶)(2nd ‘𝐼) ∧ ∀𝑥 ∈ (Base‘𝐷)∀𝑦 ∈ (Base‘𝐷)(𝑥(2nd ‘𝐼)𝑦):(𝑥(Hom ‘𝐷)𝑦)–1-1-onto→(((1st ‘𝐼)‘𝑥)(Hom ‘𝐶)((1st ‘𝐼)‘𝑦))))
59	40, 57, 58	sylanbrc 592	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (1st ‘𝐼)((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶))(2nd ‘𝐼))
60		df-br 5102	. . 3 ⊢ ((1st ‘𝐼)((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶))(2nd ‘𝐼) ↔ ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩ ∈ ((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶)))
61	59, 60	sylib 220	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩ ∈ ((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶)))
62	9, 61	eqeltrd 2863	1 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → 𝐼 ∈ ((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶)))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1561   ∈ wcel 2143  ∀wral 3077  Vcvv 3455   ∩ cin 3904   ⊆ wss 3905  ⟨cop 4589   class class class wbr 5101   I cid 5542   × cxp 5646   ↾ cres 5650  Rel wrel 5653  –1-1-onto→wf1o 6521  ‘cfv 6522  (class class class)co 7397  1^st c1st 7969  2^nd c2nd 7970  Basecbs 17246   ↾_s cress 17267  Hom chom 17298  Catccat 17697  Hom_f chomf 17699  comp_fccomf 17700   ↾_cat cresc 17842  Subcatcsubc 17843   Func cfunc 17888  id_funccidfu 17889   Full cful 17938   Faith cfth 17939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7719  ax-cnex 11130  ax-resscn 11131  ax-1cn 11132  ax-icn 11133  ax-addcl 11134  ax-addrcl 11135  ax-mulcl 11136  ax-mulrcl 11137  ax-mulcom 11138  ax-addass 11139  ax-mulass 11140  ax-distr 11141  ax-i2m1 11142  ax-1ne0 11143  ax-1rid 11144  ax-rnegex 11145  ax-rrecex 11146  ax-cnre 11147  ax-pre-lttri 11148  ax-pre-lttrn 11149  ax-pre-ltadd 11150  ax-pre-mulgt0 11151

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6289  df-ord 6350  df-on 6351  df-lim 6352  df-suc 6353  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-f1 6527  df-fo 6528  df-f1o 6529  df-fv 6530  df-riota 7354  df-ov 7400  df-oprab 7401  df-mpo 7402  df-om 7848  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8382  df-er 8679  df-map 8811  df-pm 8812  df-ixp 8881  df-en 8929  df-dom 8930  df-sdom 8931  df-pnf 11219  df-mnf 11220  df-xr 11221  df-ltxr 11222  df-le 11223  df-sub 11417  df-neg 11418  df-nn 12212  df-2 12281  df-3 12282  df-4 12283  df-5 12284  df-6 12285  df-7 12286  df-8 12287  df-9 12288  df-n0 12483  df-z 12570  df-dec 12690  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17247  df-ress 17268  df-hom 17311  df-cco 17312  df-cat 17701  df-cid 17702  df-homf 17703  df-comf 17704  df-ssc 17844  df-resc 17845  df-subc 17846  df-func 17892  df-idfu 17893  df-full 17940  df-fth 17941

This theorem is referenced by: (None)