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Theorem idffth 16814
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 16743 . . 3 Rel (𝐶 Func 𝐶)
2 idffth.i . . . 4 𝐼 = (idfunc𝐶)
32idfucl 16762 . . 3 (𝐶 ∈ Cat → 𝐼 ∈ (𝐶 Func 𝐶))
4 1st2nd 7382 . . 3 ((Rel (𝐶 Func 𝐶) ∧ 𝐼 ∈ (𝐶 Func 𝐶)) → 𝐼 = ⟨(1st𝐼), (2nd𝐼)⟩)
51, 3, 4sylancr 698 . 2 (𝐶 ∈ Cat → 𝐼 = ⟨(1st𝐼), (2nd𝐼)⟩)
65, 3eqeltrrd 2840 . . . . 5 (𝐶 ∈ Cat → ⟨(1st𝐼), (2nd𝐼)⟩ ∈ (𝐶 Func 𝐶))
7 df-br 4805 . . . . 5 ((1st𝐼)(𝐶 Func 𝐶)(2nd𝐼) ↔ ⟨(1st𝐼), (2nd𝐼)⟩ ∈ (𝐶 Func 𝐶))
86, 7sylibr 224 . . . 4 (𝐶 ∈ Cat → (1st𝐼)(𝐶 Func 𝐶)(2nd𝐼))
9 f1oi 6336 . . . . . 6 ( I ↾ (𝑥(Hom ‘𝐶)𝑦)):(𝑥(Hom ‘𝐶)𝑦)–1-1-onto→(𝑥(Hom ‘𝐶)𝑦)
10 eqid 2760 . . . . . . . 8 (Base‘𝐶) = (Base‘𝐶)
11 simpl 474 . . . . . . . 8 ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
12 eqid 2760 . . . . . . . 8 (Hom ‘𝐶) = (Hom ‘𝐶)
13 simprl 811 . . . . . . . 8 ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
14 simprr 813 . . . . . . . 8 ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
152, 10, 11, 12, 13, 14idfu2nd 16758 . . . . . . 7 ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝐼)𝑦) = ( I ↾ (𝑥(Hom ‘𝐶)𝑦)))
16 eqidd 2761 . . . . . . 7 ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐶)𝑦))
172, 10, 11, 13idfu1 16761 . . . . . . . 8 ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝐼)‘𝑥) = 𝑥)
182, 10, 11, 14idfu1 16761 . . . . . . . 8 ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝐼)‘𝑦) = 𝑦)
1917, 18oveq12d 6832 . . . . . . 7 ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (((1st𝐼)‘𝑥)(Hom ‘𝐶)((1st𝐼)‘𝑦)) = (𝑥(Hom ‘𝐶)𝑦))
2015, 16, 19f1oeq123d 6295 . . . . . 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 248 . . . . 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 3109 . . . 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 16797 . . . 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 701 . . 3 (𝐶 ∈ Cat → (1st𝐼)((𝐶 Full 𝐶) ∩ (𝐶 Faith 𝐶))(2nd𝐼))
25 df-br 4805 . . 3 ((1st𝐼)((𝐶 Full 𝐶) ∩ (𝐶 Faith 𝐶))(2nd𝐼) ↔ ⟨(1st𝐼), (2nd𝐼)⟩ ∈ ((𝐶 Full 𝐶) ∩ (𝐶 Faith 𝐶)))
2624, 25sylib 208 . 2 (𝐶 ∈ Cat → ⟨(1st𝐼), (2nd𝐼)⟩ ∈ ((𝐶 Full 𝐶) ∩ (𝐶 Faith 𝐶)))
275, 26eqeltrd 2839 1 (𝐶 ∈ Cat → 𝐼 ∈ ((𝐶 Full 𝐶) ∩ (𝐶 Faith 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wral 3050  cin 3714  cop 4327   class class class wbr 4804   I cid 5173  cres 5268  Rel wrel 5271  1-1-ontowf1o 6048  cfv 6049  (class class class)co 6814  1st c1st 7332  2nd c2nd 7333  Basecbs 16079  Hom chom 16174  Catccat 16546   Func cfunc 16735  idfunccidfu 16736   Full cful 16783   Faith cfth 16784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-map 8027  df-ixp 8077  df-cat 16550  df-cid 16551  df-func 16739  df-idfu 16740  df-full 16785  df-fth 16786
This theorem is referenced by:  rescfth  16818
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