Theorem idffth 17893

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 17819	. . 3 ⊢ Rel (𝐶 Func 𝐶)
2		idffth.i	. . . 4 ⊢ 𝐼 = (idfunc‘𝐶)
3	2	idfucl 17838	. . 3 ⊢ (𝐶 ∈ Cat → 𝐼 ∈ (𝐶 Func 𝐶))
4		1st2nd 8021	. . 3 ⊢ ((Rel (𝐶 Func 𝐶) ∧ 𝐼 ∈ (𝐶 Func 𝐶)) → 𝐼 = ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩)
5	1, 3, 4	sylancr 586	. 2 ⊢ (𝐶 ∈ Cat → 𝐼 = ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩)
6	5, 3	eqeltrrd 2828	. . . . 5 ⊢ (𝐶 ∈ Cat → ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩ ∈ (𝐶 Func 𝐶))
7		df-br 5142	. . . . 5 ⊢ ((1st ‘𝐼)(𝐶 Func 𝐶)(2nd ‘𝐼) ↔ ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩ ∈ (𝐶 Func 𝐶))
8	6, 7	sylibr 233	. . . 4 ⊢ (𝐶 ∈ Cat → (1st ‘𝐼)(𝐶 Func 𝐶)(2nd ‘𝐼))
9		f1oi 6864	. . . . . 6 ⊢ ( I ↾ (𝑥(Hom ‘𝐶)𝑦)):(𝑥(Hom ‘𝐶)𝑦)–1-1-onto→(𝑥(Hom ‘𝐶)𝑦)
10		eqid 2726	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
11		simpl 482	. . . . . . . 8 ⊢ ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
12		eqid 2726	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
13		simprl 768	. . . . . . . 8 ⊢ ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
14		simprr 770	. . . . . . . 8 ⊢ ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
15	2, 10, 11, 12, 13, 14	idfu2nd 17834	. . . . . . 7 ⊢ ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘𝐼)𝑦) = ( I ↾ (𝑥(Hom ‘𝐶)𝑦)))
16		eqidd 2727	. . . . . . 7 ⊢ ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐶)𝑦))
17	2, 10, 11, 13	idfu1 17837	. . . . . . . 8 ⊢ ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘𝐼)‘𝑥) = 𝑥)
18	2, 10, 11, 14	idfu1 17837	. . . . . . . 8 ⊢ ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘𝐼)‘𝑦) = 𝑦)
19	17, 18	oveq12d 7422	. . . . . . 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 6820	. . . . . 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 3194	. . . 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 17876	. . . 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 582	. . 3 ⊢ (𝐶 ∈ Cat → (1st ‘𝐼)((𝐶 Full 𝐶) ∩ (𝐶 Faith 𝐶))(2nd ‘𝐼))
25		df-br 5142	. . 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 217	. 2 ⊢ (𝐶 ∈ Cat → ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩ ∈ ((𝐶 Full 𝐶) ∩ (𝐶 Faith 𝐶)))
27	5, 26	eqeltrd 2827	1 ⊢ (𝐶 ∈ Cat → 𝐼 ∈ ((𝐶 Full 𝐶) ∩ (𝐶 Faith 𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ∀wral 3055   ∩ cin 3942  ⟨cop 4629   class class class wbr 5141   I cid 5566   ↾ cres 5671  Rel wrel 5674  –1-1-onto→wf1o 6535  ‘cfv 6536  (class class class)co 7404  1^st c1st 7969  2^nd c2nd 7970  Basecbs 17151  Hom chom 17215  Catccat 17615   Func cfunc 17811  id_funccidfu 17812   Full cful 17862   Faith cfth 17863

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7971  df-2nd 7972  df-map 8821  df-ixp 8891  df-cat 17619  df-cid 17620  df-func 17815  df-idfu 17816  df-full 17864  df-fth 17865

This theorem is referenced by:  rescfth  17897