Theorem idffth 18027

Description: The identity functor is a fully faithful functor. (Contributed by Mario Carneiro, 27-Jan-2017.)

Hypothesis

Ref	Expression
idffth.i	⊢ 𝐼 = (idfunc‘𝐶)

Assertion

Ref	Expression
idffth	⊢ (𝐶 ∈ Cat → 𝐼 ∈ ((𝐶 Full 𝐶) ∩ (𝐶 Faith 𝐶)))

Proof of Theorem idffth

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

Step	Hyp	Ref	Expression
1		relfunc 17954	. . 3 ⊢ Rel (𝐶 Func 𝐶)
2		idffth.i	. . . 4 ⊢ 𝐼 = (idfunc‘𝐶)
3	2	idfucl 17973	. . 3 ⊢ (𝐶 ∈ Cat → 𝐼 ∈ (𝐶 Func 𝐶))
4		1st2nd 8037	. . 3 ⊢ ((Rel (𝐶 Func 𝐶) ∧ 𝐼 ∈ (𝐶 Func 𝐶)) → 𝐼 = ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩)
5	1, 3, 4	sylancr 599	. 2 ⊢ (𝐶 ∈ Cat → 𝐼 = ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩)
6	5, 3	eqeltrrd 2861	. . . . 5 ⊢ (𝐶 ∈ Cat → ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩ ∈ (𝐶 Func 𝐶))
7		df-br 5104	. . . . 5 ⊢ ((1st ‘𝐼)(𝐶 Func 𝐶)(2nd ‘𝐼) ↔ ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩ ∈ (𝐶 Func 𝐶))
8	6, 7	sylibr 237	. . . 4 ⊢ (𝐶 ∈ Cat → (1st ‘𝐼)(𝐶 Func 𝐶)(2nd ‘𝐼))
9		f1oi 6857	. . . . . 6 ⊢ ( I ↾ (𝑥(Hom ‘𝐶)𝑦)):(𝑥(Hom ‘𝐶)𝑦)–1-1-onto→(𝑥(Hom ‘𝐶)𝑦)
10		eqid 2760	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
11		simpl 488	. . . . . . . 8 ⊢ ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
12		eqid 2760	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
13		simprl 783	. . . . . . . 8 ⊢ ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
14		simprr 785	. . . . . . . 8 ⊢ ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
15	2, 10, 11, 12, 13, 14	idfu2nd 17969	. . . . . . 7 ⊢ ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘𝐼)𝑦) = ( I ↾ (𝑥(Hom ‘𝐶)𝑦)))
16		eqidd 2761	. . . . . . 7 ⊢ ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐶)𝑦))
17	2, 10, 11, 13	idfu1 17972	. . . . . . . 8 ⊢ ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘𝐼)‘𝑥) = 𝑥)
18	2, 10, 11, 14	idfu1 17972	. . . . . . . 8 ⊢ ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘𝐼)‘𝑦) = 𝑦)
19	17, 18	oveq12d 7432	. . . . . . 7 ⊢ ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (((1st ‘𝐼)‘𝑥)(Hom ‘𝐶)((1st ‘𝐼)‘𝑦)) = (𝑥(Hom ‘𝐶)𝑦))
20	15, 16, 19	f1oeq123d 6812	. . . . . 6 ⊢ ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((𝑥(2nd ‘𝐼)𝑦):(𝑥(Hom ‘𝐶)𝑦)–1-1-onto→(((1st ‘𝐼)‘𝑥)(Hom ‘𝐶)((1st ‘𝐼)‘𝑦)) ↔ ( I ↾ (𝑥(Hom ‘𝐶)𝑦)):(𝑥(Hom ‘𝐶)𝑦)–1-1-onto→(𝑥(Hom ‘𝐶)𝑦)))
21	9, 20	mpbiri 261	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘𝐼)𝑦):(𝑥(Hom ‘𝐶)𝑦)–1-1-onto→(((1st ‘𝐼)‘𝑥)(Hom ‘𝐶)((1st ‘𝐼)‘𝑦)))
22	21	ralrimivva 3205	. . . 4 ⊢ (𝐶 ∈ Cat → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(2nd ‘𝐼)𝑦):(𝑥(Hom ‘𝐶)𝑦)–1-1-onto→(((1st ‘𝐼)‘𝑥)(Hom ‘𝐶)((1st ‘𝐼)‘𝑦)))
23	10, 12, 12	isffth2 18010	. . . 4 ⊢ ((1st ‘𝐼)((𝐶 Full 𝐶) ∩ (𝐶 Faith 𝐶))(2nd ‘𝐼) ↔ ((1st ‘𝐼)(𝐶 Func 𝐶)(2nd ‘𝐼) ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(2nd ‘𝐼)𝑦):(𝑥(Hom ‘𝐶)𝑦)–1-1-onto→(((1st ‘𝐼)‘𝑥)(Hom ‘𝐶)((1st ‘𝐼)‘𝑦))))
24	8, 22, 23	sylanbrc 595	. . 3 ⊢ (𝐶 ∈ Cat → (1st ‘𝐼)((𝐶 Full 𝐶) ∩ (𝐶 Faith 𝐶))(2nd ‘𝐼))
25		df-br 5104	. . 3 ⊢ ((1st ‘𝐼)((𝐶 Full 𝐶) ∩ (𝐶 Faith 𝐶))(2nd ‘𝐼) ↔ ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩ ∈ ((𝐶 Full 𝐶) ∩ (𝐶 Faith 𝐶)))
26	24, 25	sylib 221	. 2 ⊢ (𝐶 ∈ Cat → ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩ ∈ ((𝐶 Full 𝐶) ∩ (𝐶 Faith 𝐶)))
27	5, 26	eqeltrd 2860	1 ⊢ (𝐶 ∈ Cat → 𝐼 ∈ ((𝐶 Full 𝐶) ∩ (𝐶 Faith 𝐶)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076   ∩ cin 3898  ⟨cop 4590   class class class wbr 5103   I cid 5549   ↾ cres 5657  Rel wrel 5660  –1-1-onto→wf1o 6532  ‘cfv 6533  (class class class)co 7414  1^st c1st 7985  2^nd c2nd 7986  Basecbs 17304  Hom chom 17356  Catccat 17755   Func cfunc 17946  id_funccidfu 17947   Full cful 17996   Faith cfth 17997

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-map 8831  df-ixp 8908  df-cat 17759  df-cid 17760  df-func 17950  df-idfu 17951  df-full 17998  df-fth 17999

This theorem is used by:  rescfth  18031