Theorem ressffth 17570

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 17493	. . 3 ⊢ Rel (𝐷 Func 𝐷)
2		ressffth.d	. . . . 5 ⊢ 𝐷 = (𝐶 ↾s 𝑆)
3		resscat 17483	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (𝐶 ↾s 𝑆) ∈ Cat)
4	2, 3	eqeltrid 2843	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → 𝐷 ∈ Cat)
5		ressffth.i	. . . . 5 ⊢ 𝐼 = (idfunc‘𝐷)
6	5	idfucl 17512	. . . 4 ⊢ (𝐷 ∈ Cat → 𝐼 ∈ (𝐷 Func 𝐷))
7	4, 6	syl 17	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → 𝐼 ∈ (𝐷 Func 𝐷))
8		1st2nd 7853	. . 3 ⊢ ((Rel (𝐷 Func 𝐷) ∧ 𝐼 ∈ (𝐷 Func 𝐷)) → 𝐼 = ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩)
9	1, 7, 8	sylancr 586	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → 𝐼 = ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩)
10		eqidd 2739	. . . . . . . . 9 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (Homf ‘𝐷) = (Homf ‘𝐷))
11		eqidd 2739	. . . . . . . . 9 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (compf‘𝐷) = (compf‘𝐷))
12		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐶) = (Base‘𝐶)
13	12	ressinbas 16881	. . . . . . . . . . . . 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 2790	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → 𝐷 = (𝐶 ↾s (𝑆 ∩ (Base‘𝐶))))
16	15	fveq2d 6760	. . . . . . . . . 10 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (Homf ‘𝐷) = (Homf ‘(𝐶 ↾s (𝑆 ∩ (Base‘𝐶)))))
17		eqid 2738	. . . . . . . . . . . 12 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
18		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → 𝐶 ∈ Cat)
19		inss2 4160	. . . . . . . . . . . . 13 ⊢ (𝑆 ∩ (Base‘𝐶)) ⊆ (Base‘𝐶)
20	19	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (𝑆 ∩ (Base‘𝐶)) ⊆ (Base‘𝐶))
21		eqid 2738	. . . . . . . . . . . 12 ⊢ (𝐶 ↾s (𝑆 ∩ (Base‘𝐶))) = (𝐶 ↾s (𝑆 ∩ (Base‘𝐶)))
22		eqid 2738	. . . . . . . . . . . 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 17482	. . . . . . . . . . 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 2778	. . . . . . . . 9 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (Homf ‘𝐷) = (Homf ‘(𝐶 ↾cat ((Homf ‘𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))))
26	15	fveq2d 6760	. . . . . . . . . 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 2778	. . . . . . . . 9 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (compf‘𝐷) = (compf‘(𝐶 ↾cat ((Homf ‘𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))))
29	2	ovexi 7289	. . . . . . . . . 10 ⊢ 𝐷 ∈ V
30	29	a1i 11	. . . . . . . . 9 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → 𝐷 ∈ V)
31		ovexd 7290	. . . . . . . . 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 17532	. . . . . . . 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 17481	. . . . . . . . 9 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → ((Homf ‘𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))) ∈ (Subcat‘𝐶))
34		funcres2 17529	. . . . . . . . 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 3955	. . . . . . 7 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (𝐷 Func 𝐷) ⊆ (𝐷 Func 𝐶))
37	36, 7	sseldd 3918	. . . . . 6 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → 𝐼 ∈ (𝐷 Func 𝐶))
38	9, 37	eqeltrrd 2840	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩ ∈ (𝐷 Func 𝐶))
39		df-br 5071	. . . . 5 ⊢ ((1st ‘𝐼)(𝐷 Func 𝐶)(2nd ‘𝐼) ↔ ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩ ∈ (𝐷 Func 𝐶))
40	38, 39	sylibr 233	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (1st ‘𝐼)(𝐷 Func 𝐶)(2nd ‘𝐼))
41		f1oi 6737	. . . . . 6 ⊢ ( I ↾ (𝑥(Hom ‘𝐷)𝑦)):(𝑥(Hom ‘𝐷)𝑦)–1-1-onto→(𝑥(Hom ‘𝐷)𝑦)
42		eqid 2738	. . . . . . . 8 ⊢ (Base‘𝐷) = (Base‘𝐷)
43	4	adantr 480	. . . . . . . 8 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐷 ∈ Cat)
44		eqid 2738	. . . . . . . 8 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
45		simprl 767	. . . . . . . 8 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑥 ∈ (Base‘𝐷))
46		simprr 769	. . . . . . . 8 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑦 ∈ (Base‘𝐷))
47	5, 42, 43, 44, 45, 46	idfu2nd 17508	. . . . . . 7 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑥(2nd ‘𝐼)𝑦) = ( I ↾ (𝑥(Hom ‘𝐷)𝑦)))
48		eqidd 2739	. . . . . . 7 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
49		eqid 2738	. . . . . . . . . 10 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
50	2, 49	resshom 17046	. . . . . . . . 9 ⊢ (𝑆 ∈ 𝑉 → (Hom ‘𝐶) = (Hom ‘𝐷))
51	50	ad2antlr 723	. . . . . . . 8 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (Hom ‘𝐶) = (Hom ‘𝐷))
52	5, 42, 43, 45	idfu1 17511	. . . . . . . 8 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → ((1st ‘𝐼)‘𝑥) = 𝑥)
53	5, 42, 43, 46	idfu1 17511	. . . . . . . 8 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → ((1st ‘𝐼)‘𝑦) = 𝑦)
54	51, 52, 53	oveq123d 7276	. . . . . . 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 6694	. . . . . 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 257	. . . . 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 3114	. . . 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 17548	. . . 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 582	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (1st ‘𝐼)((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶))(2nd ‘𝐼))
60		df-br 5071	. . 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 217	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩ ∈ ((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶)))
62	9, 61	eqeltrd 2839	1 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → 𝐼 ∈ ((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  Vcvv 3422   ∩ cin 3882   ⊆ wss 3883  ⟨cop 4564   class class class wbr 5070   I cid 5479   × cxp 5578   ↾ cres 5582  Rel wrel 5585  –1-1-onto→wf1o 6417  ‘cfv 6418  (class class class)co 7255  1^st c1st 7802  2^nd c2nd 7803  Basecbs 16840   ↾_s cress 16867  Hom chom 16899  Catccat 17290  Hom_f chomf 17292  comp_fccomf 17293   ↾_cat cresc 17437  Subcatcsubc 17438   Func cfunc 17485  id_funccidfu 17486   Full cful 17534   Faith cfth 17535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-er 8456  df-map 8575  df-pm 8576  df-ixp 8644  df-en 8692  df-dom 8693  df-sdom 8694  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-7 11971  df-8 11972  df-9 11973  df-n0 12164  df-z 12250  df-dec 12367  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-hom 16912  df-cco 16913  df-cat 17294  df-cid 17295  df-homf 17296  df-comf 17297  df-ssc 17439  df-resc 17440  df-subc 17441  df-func 17489  df-idfu 17490  df-full 17536  df-fth 17537

This theorem is referenced by: (None)