Theorem ressffth 17902

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 17824	. . 3 ⊢ Rel (𝐷 Func 𝐷)
2		ressffth.d	. . . . 5 ⊢ 𝐷 = (𝐶 ↾s 𝑆)
3		resscat 17814	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (𝐶 ↾s 𝑆) ∈ Cat)
4	2, 3	eqeltrid 2832	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → 𝐷 ∈ Cat)
5		ressffth.i	. . . . 5 ⊢ 𝐼 = (idfunc‘𝐷)
6	5	idfucl 17843	. . . 4 ⊢ (𝐷 ∈ Cat → 𝐼 ∈ (𝐷 Func 𝐷))
7	4, 6	syl 17	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → 𝐼 ∈ (𝐷 Func 𝐷))
8		1st2nd 8018	. . 3 ⊢ ((Rel (𝐷 Func 𝐷) ∧ 𝐼 ∈ (𝐷 Func 𝐷)) → 𝐼 = ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩)
9	1, 7, 8	sylancr 587	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → 𝐼 = ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩)
10		eqidd 2730	. . . . . . . . 9 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (Homf ‘𝐷) = (Homf ‘𝐷))
11		eqidd 2730	. . . . . . . . 9 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (compf‘𝐷) = (compf‘𝐷))
12		eqid 2729	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐶) = (Base‘𝐶)
13	12	ressinbas 17215	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ 𝑉 → (𝐶 ↾s 𝑆) = (𝐶 ↾s (𝑆 ∩ (Base‘𝐶))))
14	13	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (𝐶 ↾s 𝑆) = (𝐶 ↾s (𝑆 ∩ (Base‘𝐶))))
15	2, 14	eqtrid 2776	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → 𝐷 = (𝐶 ↾s (𝑆 ∩ (Base‘𝐶))))
16	15	fveq2d 6862	. . . . . . . . . 10 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (Homf ‘𝐷) = (Homf ‘(𝐶 ↾s (𝑆 ∩ (Base‘𝐶)))))
17		eqid 2729	. . . . . . . . . . . 12 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
18		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → 𝐶 ∈ Cat)
19		inss2 4201	. . . . . . . . . . . . 13 ⊢ (𝑆 ∩ (Base‘𝐶)) ⊆ (Base‘𝐶)
20	19	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (𝑆 ∩ (Base‘𝐶)) ⊆ (Base‘𝐶))
21		eqid 2729	. . . . . . . . . . . 12 ⊢ (𝐶 ↾s (𝑆 ∩ (Base‘𝐶))) = (𝐶 ↾s (𝑆 ∩ (Base‘𝐶)))
22		eqid 2729	. . . . . . . . . . . 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 17813	. . . . . . . . . . 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 494	. . . . . . . . . 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 2764	. . . . . . . . 9 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (Homf ‘𝐷) = (Homf ‘(𝐶 ↾cat ((Homf ‘𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))))
26	15	fveq2d 6862	. . . . . . . . . 10 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (compf‘𝐷) = (compf‘(𝐶 ↾s (𝑆 ∩ (Base‘𝐶)))))
27	23	simprd 495	. . . . . . . . . 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 2764	. . . . . . . . 9 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (compf‘𝐷) = (compf‘(𝐶 ↾cat ((Homf ‘𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))))
29	2	ovexi 7421	. . . . . . . . . 10 ⊢ 𝐷 ∈ V
30	29	a1i 11	. . . . . . . . 9 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → 𝐷 ∈ V)
31		ovexd 7422	. . . . . . . . 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 17864	. . . . . . . 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 17812	. . . . . . . . 9 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → ((Homf ‘𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))) ∈ (Subcat‘𝐶))
34		funcres2 17860	. . . . . . . . 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 3981	. . . . . . 7 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (𝐷 Func 𝐷) ⊆ (𝐷 Func 𝐶))
37	36, 7	sseldd 3947	. . . . . 6 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → 𝐼 ∈ (𝐷 Func 𝐶))
38	9, 37	eqeltrrd 2829	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩ ∈ (𝐷 Func 𝐶))
39		df-br 5108	. . . . 5 ⊢ ((1st ‘𝐼)(𝐷 Func 𝐶)(2nd ‘𝐼) ↔ ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩ ∈ (𝐷 Func 𝐶))
40	38, 39	sylibr 234	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (1st ‘𝐼)(𝐷 Func 𝐶)(2nd ‘𝐼))
41		f1oi 6838	. . . . . 6 ⊢ ( I ↾ (𝑥(Hom ‘𝐷)𝑦)):(𝑥(Hom ‘𝐷)𝑦)–1-1-onto→(𝑥(Hom ‘𝐷)𝑦)
42		eqid 2729	. . . . . . . 8 ⊢ (Base‘𝐷) = (Base‘𝐷)
43	4	adantr 480	. . . . . . . 8 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐷 ∈ Cat)
44		eqid 2729	. . . . . . . 8 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
45		simprl 770	. . . . . . . 8 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑥 ∈ (Base‘𝐷))
46		simprr 772	. . . . . . . 8 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑦 ∈ (Base‘𝐷))
47	5, 42, 43, 44, 45, 46	idfu2nd 17839	. . . . . . 7 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑥(2nd ‘𝐼)𝑦) = ( I ↾ (𝑥(Hom ‘𝐷)𝑦)))
48		eqidd 2730	. . . . . . 7 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
49		eqid 2729	. . . . . . . . . 10 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
50	2, 49	resshom 17381	. . . . . . . . 9 ⊢ (𝑆 ∈ 𝑉 → (Hom ‘𝐶) = (Hom ‘𝐷))
51	50	ad2antlr 727	. . . . . . . 8 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (Hom ‘𝐶) = (Hom ‘𝐷))
52	5, 42, 43, 45	idfu1 17842	. . . . . . . 8 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → ((1st ‘𝐼)‘𝑥) = 𝑥)
53	5, 42, 43, 46	idfu1 17842	. . . . . . . 8 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → ((1st ‘𝐼)‘𝑦) = 𝑦)
54	51, 52, 53	oveq123d 7408	. . . . . . 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 6794	. . . . . 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 258	. . . . 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 3180	. . . 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 17880	. . . 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 583	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (1st ‘𝐼)((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶))(2nd ‘𝐼))
60		df-br 5108	. . 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 218	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩ ∈ ((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶)))
62	9, 61	eqeltrd 2828	1 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → 𝐼 ∈ ((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3447   ∩ cin 3913   ⊆ wss 3914  ⟨cop 4595   class class class wbr 5107   I cid 5532   × cxp 5636   ↾ cres 5640  Rel wrel 5643  –1-1-onto→wf1o 6510  ‘cfv 6511  (class class class)co 7387  1^st c1st 7966  2^nd c2nd 7967  Basecbs 17179   ↾_s cress 17200  Hom chom 17231  Catccat 17625  Hom_f chomf 17627  comp_fccomf 17628   ↾_cat cresc 17770  Subcatcsubc 17771   Func cfunc 17816  id_funccidfu 17817   Full cful 17866   Faith cfth 17867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-er 8671  df-map 8801  df-pm 8802  df-ixp 8871  df-en 8919  df-dom 8920  df-sdom 8921  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-z 12530  df-dec 12650  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-hom 17244  df-cco 17245  df-cat 17629  df-cid 17630  df-homf 17631  df-comf 17632  df-ssc 17772  df-resc 17773  df-subc 17774  df-func 17820  df-idfu 17821  df-full 17868  df-fth 17869

This theorem is referenced by: (None)