Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  idffth Structured version   Visualization version   GIF version

Theorem idffth 16416
 Description: The identity functor is a fully faithful functor. (Contributed by Mario Carneiro, 27-Jan-2017.)
Hypothesis
Ref Expression
idffth.i 𝐼 = (idfunc𝐶)
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.
StepHypRef Expression
1 relfunc 16345 . . 3 Rel (𝐶 Func 𝐶)
2 idffth.i . . . 4 𝐼 = (idfunc𝐶)
32idfucl 16364 . . 3 (𝐶 ∈ Cat → 𝐼 ∈ (𝐶 Func 𝐶))
4 1st2nd 7105 . . 3 ((Rel (𝐶 Func 𝐶) ∧ 𝐼 ∈ (𝐶 Func 𝐶)) → 𝐼 = ⟨(1st𝐼), (2nd𝐼)⟩)
51, 3, 4sylancr 694 . 2 (𝐶 ∈ Cat → 𝐼 = ⟨(1st𝐼), (2nd𝐼)⟩)
65, 3eqeltrrd 2689 . . . . 5 (𝐶 ∈ Cat → ⟨(1st𝐼), (2nd𝐼)⟩ ∈ (𝐶 Func 𝐶))
7 df-br 4584 . . . . 5 ((1st𝐼)(𝐶 Func 𝐶)(2nd𝐼) ↔ ⟨(1st𝐼), (2nd𝐼)⟩ ∈ (𝐶 Func 𝐶))
86, 7sylibr 223 . . . 4 (𝐶 ∈ Cat → (1st𝐼)(𝐶 Func 𝐶)(2nd𝐼))
9 f1oi 6086 . . . . . 6 ( I ↾ (𝑥(Hom ‘𝐶)𝑦)):(𝑥(Hom ‘𝐶)𝑦)–1-1-onto→(𝑥(Hom ‘𝐶)𝑦)
10 eqid 2610 . . . . . . . 8 (Base‘𝐶) = (Base‘𝐶)
11 simpl 472 . . . . . . . 8 ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
12 eqid 2610 . . . . . . . 8 (Hom ‘𝐶) = (Hom ‘𝐶)
13 simprl 790 . . . . . . . 8 ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
14 simprr 792 . . . . . . . 8 ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
152, 10, 11, 12, 13, 14idfu2nd 16360 . . . . . . 7 ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝐼)𝑦) = ( I ↾ (𝑥(Hom ‘𝐶)𝑦)))
16 eqidd 2611 . . . . . . 7 ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐶)𝑦))
172, 10, 11, 13idfu1 16363 . . . . . . . 8 ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝐼)‘𝑥) = 𝑥)
182, 10, 11, 14idfu1 16363 . . . . . . . 8 ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝐼)‘𝑦) = 𝑦)
1917, 18oveq12d 6567 . . . . . . 7 ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (((1st𝐼)‘𝑥)(Hom ‘𝐶)((1st𝐼)‘𝑦)) = (𝑥(Hom ‘𝐶)𝑦))
2015, 16, 19f1oeq123d 6046 . . . . . 6 ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((𝑥(2nd𝐼)𝑦):(𝑥(Hom ‘𝐶)𝑦)–1-1-onto→(((1st𝐼)‘𝑥)(Hom ‘𝐶)((1st𝐼)‘𝑦)) ↔ ( I ↾ (𝑥(Hom ‘𝐶)𝑦)):(𝑥(Hom ‘𝐶)𝑦)–1-1-onto→(𝑥(Hom ‘𝐶)𝑦)))
219, 20mpbiri 247 . . . . 5 ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝐼)𝑦):(𝑥(Hom ‘𝐶)𝑦)–1-1-onto→(((1st𝐼)‘𝑥)(Hom ‘𝐶)((1st𝐼)‘𝑦)))
2221ralrimivva 2954 . . . 4 (𝐶 ∈ Cat → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(2nd𝐼)𝑦):(𝑥(Hom ‘𝐶)𝑦)–1-1-onto→(((1st𝐼)‘𝑥)(Hom ‘𝐶)((1st𝐼)‘𝑦)))
2310, 12, 12isffth2 16399 . . . 4 ((1st𝐼)((𝐶 Full 𝐶) ∩ (𝐶 Faith 𝐶))(2nd𝐼) ↔ ((1st𝐼)(𝐶 Func 𝐶)(2nd𝐼) ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(2nd𝐼)𝑦):(𝑥(Hom ‘𝐶)𝑦)–1-1-onto→(((1st𝐼)‘𝑥)(Hom ‘𝐶)((1st𝐼)‘𝑦))))
248, 22, 23sylanbrc 695 . . 3 (𝐶 ∈ Cat → (1st𝐼)((𝐶 Full 𝐶) ∩ (𝐶 Faith 𝐶))(2nd𝐼))
25 df-br 4584 . . 3 ((1st𝐼)((𝐶 Full 𝐶) ∩ (𝐶 Faith 𝐶))(2nd𝐼) ↔ ⟨(1st𝐼), (2nd𝐼)⟩ ∈ ((𝐶 Full 𝐶) ∩ (𝐶 Faith 𝐶)))
2624, 25sylib 207 . 2 (𝐶 ∈ Cat → ⟨(1st𝐼), (2nd𝐼)⟩ ∈ ((𝐶 Full 𝐶) ∩ (𝐶 Faith 𝐶)))
275, 26eqeltrd 2688 1 (𝐶 ∈ Cat → 𝐼 ∈ ((𝐶 Full 𝐶) ∩ (𝐶 Faith 𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∩ cin 3539  ⟨cop 4131   class class class wbr 4583   I cid 4948   ↾ cres 5040  Rel wrel 5043  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  Basecbs 15695  Hom chom 15779  Catccat 16148   Func cfunc 16337  idfunccidfu 16338   Full cful 16385   Faith cfth 16386 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-ixp 7795  df-cat 16152  df-cid 16153  df-func 16341  df-idfu 16342  df-full 16387  df-fth 16388 This theorem is referenced by:  rescfth  16420
 Copyright terms: Public domain W3C validator