Theorem idffth 17064

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 16993	. . 3 ⊢ Rel (𝐶 Func 𝐶)
2		idffth.i	. . . 4 ⊢ 𝐼 = (idfunc‘𝐶)
3	2	idfucl 17012	. . 3 ⊢ (𝐶 ∈ Cat → 𝐼 ∈ (𝐶 Func 𝐶))
4		1st2nd 7552	. . 3 ⊢ ((Rel (𝐶 Func 𝐶) ∧ 𝐼 ∈ (𝐶 Func 𝐶)) → 𝐼 = ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩)
5	1, 3, 4	sylancr 578	. 2 ⊢ (𝐶 ∈ Cat → 𝐼 = ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩)
6	5, 3	eqeltrrd 2867	. . . . 5 ⊢ (𝐶 ∈ Cat → ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩ ∈ (𝐶 Func 𝐶))
7		df-br 4931	. . . . 5 ⊢ ((1st ‘𝐼)(𝐶 Func 𝐶)(2nd ‘𝐼) ↔ ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩ ∈ (𝐶 Func 𝐶))
8	6, 7	sylibr 226	. . . 4 ⊢ (𝐶 ∈ Cat → (1st ‘𝐼)(𝐶 Func 𝐶)(2nd ‘𝐼))
9		f1oi 6483	. . . . . 6 ⊢ ( I ↾ (𝑥(Hom ‘𝐶)𝑦)):(𝑥(Hom ‘𝐶)𝑦)–1-1-onto→(𝑥(Hom ‘𝐶)𝑦)
10		eqid 2778	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
11		simpl 475	. . . . . . . 8 ⊢ ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
12		eqid 2778	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
13		simprl 758	. . . . . . . 8 ⊢ ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
14		simprr 760	. . . . . . . 8 ⊢ ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
15	2, 10, 11, 12, 13, 14	idfu2nd 17008	. . . . . . 7 ⊢ ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘𝐼)𝑦) = ( I ↾ (𝑥(Hom ‘𝐶)𝑦)))
16		eqidd 2779	. . . . . . 7 ⊢ ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐶)𝑦))
17	2, 10, 11, 13	idfu1 17011	. . . . . . . 8 ⊢ ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘𝐼)‘𝑥) = 𝑥)
18	2, 10, 11, 14	idfu1 17011	. . . . . . . 8 ⊢ ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘𝐼)‘𝑦) = 𝑦)
19	17, 18	oveq12d 6996	. . . . . . 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 6441	. . . . . 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 250	. . . . 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 3141	. . . 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 17047	. . . 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 575	. . 3 ⊢ (𝐶 ∈ Cat → (1st ‘𝐼)((𝐶 Full 𝐶) ∩ (𝐶 Faith 𝐶))(2nd ‘𝐼))
25		df-br 4931	. . 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 210	. 2 ⊢ (𝐶 ∈ Cat → ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩ ∈ ((𝐶 Full 𝐶) ∩ (𝐶 Faith 𝐶)))
27	5, 26	eqeltrd 2866	1 ⊢ (𝐶 ∈ Cat → 𝐼 ∈ ((𝐶 Full 𝐶) ∩ (𝐶 Faith 𝐶)))

Syntax hints:   → wi 4   ∧ wa 387   = wceq 1507   ∈ wcel 2050  ∀wral 3088   ∩ cin 3830  ⟨cop 4448   class class class wbr 4930   I cid 5312   ↾ cres 5410  Rel wrel 5413  –1-1-onto→wf1o 6189  ‘cfv 6190  (class class class)co 6978  1^st c1st 7501  2^nd c2nd 7502  Basecbs 16342  Hom chom 16435  Catccat 16796   Func cfunc 16985  id_funccidfu 16986   Full cful 17033   Faith cfth 17034

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5050  ax-sep 5061  ax-nul 5068  ax-pow 5120  ax-pr 5187  ax-un 7281

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3417  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-if 4352  df-pw 4425  df-sn 4443  df-pr 4445  df-op 4449  df-uni 4714  df-iun 4795  df-br 4931  df-opab 4993  df-mpt 5010  df-id 5313  df-xp 5414  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-rn 5419  df-res 5420  df-ima 5421  df-iota 6154  df-fun 6192  df-fn 6193  df-f 6194  df-f1 6195  df-fo 6196  df-f1o 6197  df-fv 6198  df-riota 6939  df-ov 6981  df-oprab 6982  df-mpo 6983  df-1st 7503  df-2nd 7504  df-map 8210  df-ixp 8262  df-cat 16800  df-cid 16801  df-func 16989  df-idfu 16990  df-full 17035  df-fth 17036

This theorem is referenced by:  rescfth  17068