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Theorem idffth 16359
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 16288 . . 3 Rel (𝐶 Func 𝐶)
2 idffth.i . . . 4 𝐼 = (idfunc𝐶)
32idfucl 16307 . . 3 (𝐶 ∈ Cat → 𝐼 ∈ (𝐶 Func 𝐶))
4 1st2nd 7079 . . 3 ((Rel (𝐶 Func 𝐶) ∧ 𝐼 ∈ (𝐶 Func 𝐶)) → 𝐼 = ⟨(1st𝐼), (2nd𝐼)⟩)
51, 3, 4sylancr 693 . 2 (𝐶 ∈ Cat → 𝐼 = ⟨(1st𝐼), (2nd𝐼)⟩)
65, 3eqeltrrd 2685 . . . . 5 (𝐶 ∈ Cat → ⟨(1st𝐼), (2nd𝐼)⟩ ∈ (𝐶 Func 𝐶))
7 df-br 4575 . . . . 5 ((1st𝐼)(𝐶 Func 𝐶)(2nd𝐼) ↔ ⟨(1st𝐼), (2nd𝐼)⟩ ∈ (𝐶 Func 𝐶))
86, 7sylibr 222 . . . 4 (𝐶 ∈ Cat → (1st𝐼)(𝐶 Func 𝐶)(2nd𝐼))
9 f1oi 6068 . . . . . 6 ( I ↾ (𝑥(Hom ‘𝐶)𝑦)):(𝑥(Hom ‘𝐶)𝑦)–1-1-onto→(𝑥(Hom ‘𝐶)𝑦)
10 eqid 2606 . . . . . . . 8 (Base‘𝐶) = (Base‘𝐶)
11 simpl 471 . . . . . . . 8 ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
12 eqid 2606 . . . . . . . 8 (Hom ‘𝐶) = (Hom ‘𝐶)
13 simprl 789 . . . . . . . 8 ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
14 simprr 791 . . . . . . . 8 ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
152, 10, 11, 12, 13, 14idfu2nd 16303 . . . . . . 7 ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝐼)𝑦) = ( I ↾ (𝑥(Hom ‘𝐶)𝑦)))
16 eqidd 2607 . . . . . . 7 ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐶)𝑦))
172, 10, 11, 13idfu1 16306 . . . . . . . 8 ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝐼)‘𝑥) = 𝑥)
182, 10, 11, 14idfu1 16306 . . . . . . . 8 ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝐼)‘𝑦) = 𝑦)
1917, 18oveq12d 6542 . . . . . . 7 ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (((1st𝐼)‘𝑥)(Hom ‘𝐶)((1st𝐼)‘𝑦)) = (𝑥(Hom ‘𝐶)𝑦))
2015, 16, 19f1oeq123d 6028 . . . . . 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 246 . . . . 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 2950 . . . 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 16342 . . . 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 694 . . 3 (𝐶 ∈ Cat → (1st𝐼)((𝐶 Full 𝐶) ∩ (𝐶 Faith 𝐶))(2nd𝐼))
25 df-br 4575 . . 3 ((1st𝐼)((𝐶 Full 𝐶) ∩ (𝐶 Faith 𝐶))(2nd𝐼) ↔ ⟨(1st𝐼), (2nd𝐼)⟩ ∈ ((𝐶 Full 𝐶) ∩ (𝐶 Faith 𝐶)))
2624, 25sylib 206 . 2 (𝐶 ∈ Cat → ⟨(1st𝐼), (2nd𝐼)⟩ ∈ ((𝐶 Full 𝐶) ∩ (𝐶 Faith 𝐶)))
275, 26eqeltrd 2684 1 (𝐶 ∈ Cat → 𝐼 ∈ ((𝐶 Full 𝐶) ∩ (𝐶 Faith 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  wral 2892  cin 3535  cop 4127   class class class wbr 4574   I cid 4935  cres 5027  Rel wrel 5030  1-1-ontowf1o 5786  cfv 5787  (class class class)co 6524  1st c1st 7031  2nd c2nd 7032  Basecbs 15638  Hom chom 15722  Catccat 16091   Func cfunc 16280  idfunccidfu 16281   Full cful 16328   Faith cfth 16329
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-ral 2897  df-rex 2898  df-reu 2899  df-rmo 2900  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-op 4128  df-uni 4364  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-id 4940  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-riota 6486  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-1st 7033  df-2nd 7034  df-map 7720  df-ixp 7769  df-cat 16095  df-cid 16096  df-func 16284  df-idfu 16285  df-full 16330  df-fth 16331
This theorem is referenced by:  rescfth  16363
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