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 2841	. . . 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 7987	. . 3 ⊢ ((Rel (𝐷 Func 𝐷) ∧ 𝐼 ∈ (𝐷 Func 𝐷)) → 𝐼 = ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩)
9	1, 7, 8	sylancr 588	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → 𝐼 = ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩)
10		eqidd 2738	. . . . . . . . 9 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (Homf ‘𝐷) = (Homf ‘𝐷))
11		eqidd 2738	. . . . . . . . 9 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (compf‘𝐷) = (compf‘𝐷))
12		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐶) = (Base‘𝐶)
13	12	ressinbas 17210	. . . . . . . . . . . . 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 2784	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → 𝐷 = (𝐶 ↾s (𝑆 ∩ (Base‘𝐶))))
16	15	fveq2d 6840	. . . . . . . . . 10 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (Homf ‘𝐷) = (Homf ‘(𝐶 ↾s (𝑆 ∩ (Base‘𝐶)))))
17		eqid 2737	. . . . . . . . . . . 12 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
18		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → 𝐶 ∈ Cat)
19		inss2 4179	. . . . . . . . . . . . 13 ⊢ (𝑆 ∩ (Base‘𝐶)) ⊆ (Base‘𝐶)
20	19	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (𝑆 ∩ (Base‘𝐶)) ⊆ (Base‘𝐶))
21		eqid 2737	. . . . . . . . . . . 12 ⊢ (𝐶 ↾s (𝑆 ∩ (Base‘𝐶))) = (𝐶 ↾s (𝑆 ∩ (Base‘𝐶)))
22		eqid 2737	. . . . . . . . . . . 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 2772	. . . . . . . . 9 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (Homf ‘𝐷) = (Homf ‘(𝐶 ↾cat ((Homf ‘𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))))
26	15	fveq2d 6840	. . . . . . . . . 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 2772	. . . . . . . . 9 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (compf‘𝐷) = (compf‘(𝐶 ↾cat ((Homf ‘𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))))
29	2	ovexi 7396	. . . . . . . . . 10 ⊢ 𝐷 ∈ V
30	29	a1i 11	. . . . . . . . 9 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → 𝐷 ∈ V)
31		ovexd 7397	. . . . . . . . 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 3957	. . . . . . 7 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (𝐷 Func 𝐷) ⊆ (𝐷 Func 𝐶))
37	36, 7	sseldd 3923	. . . . . 6 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → 𝐼 ∈ (𝐷 Func 𝐶))
38	9, 37	eqeltrrd 2838	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩ ∈ (𝐷 Func 𝐶))
39		df-br 5087	. . . . 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 6814	. . . . . 6 ⊢ ( I ↾ (𝑥(Hom ‘𝐷)𝑦)):(𝑥(Hom ‘𝐷)𝑦)–1-1-onto→(𝑥(Hom ‘𝐷)𝑦)
42		eqid 2737	. . . . . . . 8 ⊢ (Base‘𝐷) = (Base‘𝐷)
43	4	adantr 480	. . . . . . . 8 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐷 ∈ Cat)
44		eqid 2737	. . . . . . . 8 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
45		simprl 771	. . . . . . . 8 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑥 ∈ (Base‘𝐷))
46		simprr 773	. . . . . . . 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 2738	. . . . . . 7 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
49		eqid 2737	. . . . . . . . . 10 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
50	2, 49	resshom 17376	. . . . . . . . 9 ⊢ (𝑆 ∈ 𝑉 → (Hom ‘𝐶) = (Hom ‘𝐷))
51	50	ad2antlr 728	. . . . . . . 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 7383	. . . . . . 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 6770	. . . . . 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 3181	. . . 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 584	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (1st ‘𝐼)((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶))(2nd ‘𝐼))
60		df-br 5087	. . 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 2837	1 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → 𝐼 ∈ ((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3430   ∩ cin 3889   ⊆ wss 3890  ⟨cop 4574   class class class wbr 5086   I cid 5520   × cxp 5624   ↾ cres 5628  Rel wrel 5631  –1-1-onto→wf1o 6493  ‘cfv 6494  (class class class)co 7362  1^st c1st 7935  2^nd c2nd 7936  Basecbs 17174   ↾_s cress 17195  Hom chom 17226  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 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-riota 7319  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7813  df-1st 7937  df-2nd 7938  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-er 8638  df-map 8770  df-pm 8771  df-ixp 8841  df-en 8889  df-dom 8890  df-sdom 8891  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-nn 12170  df-2 12239  df-3 12240  df-4 12241  df-5 12242  df-6 12243  df-7 12244  df-8 12245  df-9 12246  df-n0 12433  df-z 12520  df-dec 12640  df-sets 17129  df-slot 17147  df-ndx 17159  df-base 17175  df-ress 17196  df-hom 17239  df-cco 17240  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)