Theorem ressffth 16805

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 16729	. . 3 ⊢ Rel (𝐷 Func 𝐷)
2		ressffth.d	. . . . 5 ⊢ 𝐷 = (𝐶 ↾s 𝑆)
3		resscat 16719	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (𝐶 ↾s 𝑆) ∈ Cat)
4	2, 3	syl5eqel 2854	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → 𝐷 ∈ Cat)
5		ressffth.i	. . . . 5 ⊢ 𝐼 = (idfunc‘𝐷)
6	5	idfucl 16748	. . . 4 ⊢ (𝐷 ∈ Cat → 𝐼 ∈ (𝐷 Func 𝐷))
7	4, 6	syl 17	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → 𝐼 ∈ (𝐷 Func 𝐷))
8		1st2nd 7367	. . 3 ⊢ ((Rel (𝐷 Func 𝐷) ∧ 𝐼 ∈ (𝐷 Func 𝐷)) → 𝐼 = ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩)
9	1, 7, 8	sylancr 575	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → 𝐼 = ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩)
10		eqidd 2772	. . . . . . . . 9 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (Homf ‘𝐷) = (Homf ‘𝐷))
11		eqidd 2772	. . . . . . . . 9 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (compf‘𝐷) = (compf‘𝐷))
12		eqid 2771	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐶) = (Base‘𝐶)
13	12	ressinbas 16143	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ 𝑉 → (𝐶 ↾s 𝑆) = (𝐶 ↾s (𝑆 ∩ (Base‘𝐶))))
14	13	adantl 467	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (𝐶 ↾s 𝑆) = (𝐶 ↾s (𝑆 ∩ (Base‘𝐶))))
15	2, 14	syl5eq 2817	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → 𝐷 = (𝐶 ↾s (𝑆 ∩ (Base‘𝐶))))
16	15	fveq2d 6337	. . . . . . . . . 10 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (Homf ‘𝐷) = (Homf ‘(𝐶 ↾s (𝑆 ∩ (Base‘𝐶)))))
17		eqid 2771	. . . . . . . . . . . 12 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
18		simpl 468	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → 𝐶 ∈ Cat)
19		inss2 3982	. . . . . . . . . . . . 13 ⊢ (𝑆 ∩ (Base‘𝐶)) ⊆ (Base‘𝐶)
20	19	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (𝑆 ∩ (Base‘𝐶)) ⊆ (Base‘𝐶))
21		eqid 2771	. . . . . . . . . . . 12 ⊢ (𝐶 ↾s (𝑆 ∩ (Base‘𝐶))) = (𝐶 ↾s (𝑆 ∩ (Base‘𝐶)))
22		eqid 2771	. . . . . . . . . . . 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 16718	. . . . . . . . . . 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 482	. . . . . . . . . 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 2805	. . . . . . . . 9 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (Homf ‘𝐷) = (Homf ‘(𝐶 ↾cat ((Homf ‘𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))))
26	15	fveq2d 6337	. . . . . . . . . 10 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (compf‘𝐷) = (compf‘(𝐶 ↾s (𝑆 ∩ (Base‘𝐶)))))
27	23	simprd 483	. . . . . . . . . 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 2805	. . . . . . . . 9 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (compf‘𝐷) = (compf‘(𝐶 ↾cat ((Homf ‘𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))))
29		ovex 6827	. . . . . . . . . . 11 ⊢ (𝐶 ↾s 𝑆) ∈ V
30	2, 29	eqeltri 2846	. . . . . . . . . 10 ⊢ 𝐷 ∈ V
31	30	a1i 11	. . . . . . . . 9 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → 𝐷 ∈ V)
32		ovex 6827	. . . . . . . . . 10 ⊢ (𝐶 ↾cat ((Homf ‘𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶))))) ∈ V
33	32	a1i 11	. . . . . . . . 9 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (𝐶 ↾cat ((Homf ‘𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶))))) ∈ V)
34	10, 11, 25, 28, 31, 31, 31, 33	funcpropd 16767	. . . . . . . 8 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (𝐷 Func 𝐷) = (𝐷 Func (𝐶 ↾cat ((Homf ‘𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))))
35	12, 17, 18, 20	fullsubc 16717	. . . . . . . . 9 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → ((Homf ‘𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))) ∈ (Subcat‘𝐶))
36		funcres2 16765	. . . . . . . . 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))
37	35, 36	syl 17	. . . . . . . 8 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (𝐷 Func (𝐶 ↾cat ((Homf ‘𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))) ⊆ (𝐷 Func 𝐶))
38	34, 37	eqsstrd 3788	. . . . . . 7 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (𝐷 Func 𝐷) ⊆ (𝐷 Func 𝐶))
39	38, 7	sseldd 3753	. . . . . 6 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → 𝐼 ∈ (𝐷 Func 𝐶))
40	9, 39	eqeltrrd 2851	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩ ∈ (𝐷 Func 𝐶))
41		df-br 4788	. . . . 5 ⊢ ((1st ‘𝐼)(𝐷 Func 𝐶)(2nd ‘𝐼) ↔ ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩ ∈ (𝐷 Func 𝐶))
42	40, 41	sylibr 224	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (1st ‘𝐼)(𝐷 Func 𝐶)(2nd ‘𝐼))
43		f1oi 6316	. . . . . 6 ⊢ ( I ↾ (𝑥(Hom ‘𝐷)𝑦)):(𝑥(Hom ‘𝐷)𝑦)–1-1-onto→(𝑥(Hom ‘𝐷)𝑦)
44		eqid 2771	. . . . . . . 8 ⊢ (Base‘𝐷) = (Base‘𝐷)
45	4	adantr 466	. . . . . . . 8 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐷 ∈ Cat)
46		eqid 2771	. . . . . . . 8 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
47		simprl 754	. . . . . . . 8 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑥 ∈ (Base‘𝐷))
48		simprr 756	. . . . . . . 8 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑦 ∈ (Base‘𝐷))
49	5, 44, 45, 46, 47, 48	idfu2nd 16744	. . . . . . 7 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑥(2nd ‘𝐼)𝑦) = ( I ↾ (𝑥(Hom ‘𝐷)𝑦)))
50		eqidd 2772	. . . . . . 7 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
51		eqid 2771	. . . . . . . . . 10 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
52	2, 51	resshom 16286	. . . . . . . . 9 ⊢ (𝑆 ∈ 𝑉 → (Hom ‘𝐶) = (Hom ‘𝐷))
53	52	ad2antlr 706	. . . . . . . 8 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (Hom ‘𝐶) = (Hom ‘𝐷))
54	5, 44, 45, 47	idfu1 16747	. . . . . . . 8 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → ((1st ‘𝐼)‘𝑥) = 𝑥)
55	5, 44, 45, 48	idfu1 16747	. . . . . . . 8 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → ((1st ‘𝐼)‘𝑦) = 𝑦)
56	53, 54, 55	oveq123d 6817	. . . . . . 7 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (((1st ‘𝐼)‘𝑥)(Hom ‘𝐶)((1st ‘𝐼)‘𝑦)) = (𝑥(Hom ‘𝐷)𝑦))
57	49, 50, 56	f1oeq123d 6275	. . . . . 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)))
58	43, 57	mpbiri 248	. . . . 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)))
59	58	ralrimivva 3120	. . . 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)))
60	44, 46, 51	isffth2 16783	. . . 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))))
61	42, 59, 60	sylanbrc 572	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (1st ‘𝐼)((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶))(2nd ‘𝐼))
62		df-br 4788	. . 3 ⊢ ((1st ‘𝐼)((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶))(2nd ‘𝐼) ↔ ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩ ∈ ((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶)))
63	61, 62	sylib 208	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩ ∈ ((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶)))
64	9, 63	eqeltrd 2850	1 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → 𝐼 ∈ ((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶)))

Syntax hints:   → wi 4   ∧ wa 382   = wceq 1631   ∈ wcel 2145  ∀wral 3061  Vcvv 3351   ∩ cin 3722   ⊆ wss 3723  ⟨cop 4323   class class class wbr 4787   I cid 5157   × cxp 5248   ↾ cres 5252  Rel wrel 5255  –1-1-onto→wf1o 6029  ‘cfv 6030  (class class class)co 6796  1^st c1st 7317  2^nd c2nd 7318  Basecbs 16064   ↾_s cress 16065  Hom chom 16160  Catccat 16532  Hom_f chomf 16534  comp_fccomf 16535   ↾_cat cresc 16675  Subcatcsubc 16676   Func cfunc 16721  id_funccidfu 16722   Full cful 16769   Faith cfth 16770

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-cnex 10198  ax-resscn 10199  ax-1cn 10200  ax-icn 10201  ax-addcl 10202  ax-addrcl 10203  ax-mulcl 10204  ax-mulrcl 10205  ax-mulcom 10206  ax-addass 10207  ax-mulass 10208  ax-distr 10209  ax-i2m1 10210  ax-1ne0 10211  ax-1rid 10212  ax-rnegex 10213  ax-rrecex 10214  ax-cnre 10215  ax-pre-lttri 10216  ax-pre-lttrn 10217  ax-pre-ltadd 10218  ax-pre-mulgt0 10219

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-1st 7319  df-2nd 7320  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-er 7900  df-map 8015  df-pm 8016  df-ixp 8067  df-en 8114  df-dom 8115  df-sdom 8116  df-pnf 10282  df-mnf 10283  df-xr 10284  df-ltxr 10285  df-le 10286  df-sub 10474  df-neg 10475  df-nn 11227  df-2 11285  df-3 11286  df-4 11287  df-5 11288  df-6 11289  df-7 11290  df-8 11291  df-9 11292  df-n0 11500  df-z 11585  df-dec 11701  df-ndx 16067  df-slot 16068  df-base 16070  df-sets 16071  df-ress 16072  df-hom 16174  df-cco 16175  df-cat 16536  df-cid 16537  df-homf 16538  df-comf 16539  df-ssc 16677  df-resc 16678  df-subc 16679  df-func 16725  df-idfu 16726  df-full 16771  df-fth 16772

This theorem is referenced by: (None)