Theorem idffth 17873

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 17800	. . 3 ⊢ Rel (𝐶 Func 𝐶)
2		idffth.i	. . . 4 ⊢ 𝐼 = (idfunc‘𝐶)
3	2	idfucl 17819	. . 3 ⊢ (𝐶 ∈ Cat → 𝐼 ∈ (𝐶 Func 𝐶))
4		1st2nd 7997	. . 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 2829	. . . . 5 ⊢ (𝐶 ∈ Cat → ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩ ∈ (𝐶 Func 𝐶))
7		df-br 5103	. . . . 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 6820	. . . . . 6 ⊢ ( I ↾ (𝑥(Hom ‘𝐶)𝑦)):(𝑥(Hom ‘𝐶)𝑦)–1-1-onto→(𝑥(Hom ‘𝐶)𝑦)
10		eqid 2729	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
11		simpl 482	. . . . . . . 8 ⊢ ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
12		eqid 2729	. . . . . . . 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 17815	. . . . . . 7 ⊢ ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘𝐼)𝑦) = ( I ↾ (𝑥(Hom ‘𝐶)𝑦)))
16		eqidd 2730	. . . . . . 7 ⊢ ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐶)𝑦))
17	2, 10, 11, 13	idfu1 17818	. . . . . . . 8 ⊢ ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘𝐼)‘𝑥) = 𝑥)
18	2, 10, 11, 14	idfu1 17818	. . . . . . . 8 ⊢ ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘𝐼)‘𝑦) = 𝑦)
19	17, 18	oveq12d 7387	. . . . . . 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 6776	. . . . . 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 3178	. . . 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 17856	. . . 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 5103	. . 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 2828	1 ⊢ (𝐶 ∈ Cat → 𝐼 ∈ ((𝐶 Full 𝐶) ∩ (𝐶 Faith 𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ∩ cin 3910  ⟨cop 4591   class class class wbr 5102   I cid 5525   ↾ cres 5633  Rel wrel 5636  –1-1-onto→wf1o 6498  ‘cfv 6499  (class class class)co 7369  1^st c1st 7945  2^nd c2nd 7946  Basecbs 17155  Hom chom 17207  Catccat 17601   Func cfunc 17792  id_funccidfu 17793   Full cful 17842   Faith cfth 17843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948  df-map 8778  df-ixp 8848  df-cat 17605  df-cid 17606  df-func 17796  df-idfu 17797  df-full 17844  df-fth 17845

This theorem is referenced by:  rescfth  17877