Theorem idffth 17857

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 17784	. . 3 ⊢ Rel (𝐶 Func 𝐶)
2		idffth.i	. . . 4 ⊢ 𝐼 = (idfunc‘𝐶)
3	2	idfucl 17803	. . 3 ⊢ (𝐶 ∈ Cat → 𝐼 ∈ (𝐶 Func 𝐶))
4		1st2nd 7981	. . 3 ⊢ ((Rel (𝐶 Func 𝐶) ∧ 𝐼 ∈ (𝐶 Func 𝐶)) → 𝐼 = ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩)
5	1, 3, 4	sylancr 587	. 2 ⊢ (𝐶 ∈ Cat → 𝐼 = ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩)
6	5, 3	eqeltrrd 2835	. . . . 5 ⊢ (𝐶 ∈ Cat → ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩ ∈ (𝐶 Func 𝐶))
7		df-br 5097	. . . . 5 ⊢ ((1st ‘𝐼)(𝐶 Func 𝐶)(2nd ‘𝐼) ↔ ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩ ∈ (𝐶 Func 𝐶))
8	6, 7	sylibr 234	. . . 4 ⊢ (𝐶 ∈ Cat → (1st ‘𝐼)(𝐶 Func 𝐶)(2nd ‘𝐼))
9		f1oi 6810	. . . . . 6 ⊢ ( I ↾ (𝑥(Hom ‘𝐶)𝑦)):(𝑥(Hom ‘𝐶)𝑦)–1-1-onto→(𝑥(Hom ‘𝐶)𝑦)
10		eqid 2734	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
11		simpl 482	. . . . . . . 8 ⊢ ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
12		eqid 2734	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
13		simprl 770	. . . . . . . 8 ⊢ ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
14		simprr 772	. . . . . . . 8 ⊢ ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
15	2, 10, 11, 12, 13, 14	idfu2nd 17799	. . . . . . 7 ⊢ ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘𝐼)𝑦) = ( I ↾ (𝑥(Hom ‘𝐶)𝑦)))
16		eqidd 2735	. . . . . . 7 ⊢ ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐶)𝑦))
17	2, 10, 11, 13	idfu1 17802	. . . . . . . 8 ⊢ ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘𝐼)‘𝑥) = 𝑥)
18	2, 10, 11, 14	idfu1 17802	. . . . . . . 8 ⊢ ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘𝐼)‘𝑦) = 𝑦)
19	17, 18	oveq12d 7374	. . . . . . 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 6766	. . . . . 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 258	. . . . 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 3177	. . . 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 17840	. . . 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 583	. . 3 ⊢ (𝐶 ∈ Cat → (1st ‘𝐼)((𝐶 Full 𝐶) ∩ (𝐶 Faith 𝐶))(2nd ‘𝐼))
25		df-br 5097	. . 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 218	. 2 ⊢ (𝐶 ∈ Cat → ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩ ∈ ((𝐶 Full 𝐶) ∩ (𝐶 Faith 𝐶)))
27	5, 26	eqeltrd 2834	1 ⊢ (𝐶 ∈ Cat → 𝐼 ∈ ((𝐶 Full 𝐶) ∩ (𝐶 Faith 𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3049   ∩ cin 3898  ⟨cop 4584   class class class wbr 5096   I cid 5516   ↾ cres 5624  Rel wrel 5627  –1-1-onto→wf1o 6489  ‘cfv 6490  (class class class)co 7356  1^st c1st 7929  2^nd c2nd 7930  Basecbs 17134  Hom chom 17186  Catccat 17585   Func cfunc 17776  id_funccidfu 17777   Full cful 17826   Faith cfth 17827

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-map 8763  df-ixp 8834  df-cat 17589  df-cid 17590  df-func 17780  df-idfu 17781  df-full 17828  df-fth 17829

This theorem is referenced by:  rescfth  17861