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Theorem idffth 17880
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 17808 . . 3 Rel (𝐶 Func 𝐶)
2 idffth.i . . . 4 𝐼 = (idfunc𝐶)
32idfucl 17827 . . 3 (𝐶 ∈ Cat → 𝐼 ∈ (𝐶 Func 𝐶))
4 1st2nd 8021 . . 3 ((Rel (𝐶 Func 𝐶) ∧ 𝐼 ∈ (𝐶 Func 𝐶)) → 𝐼 = ⟨(1st𝐼), (2nd𝐼)⟩)
51, 3, 4sylancr 587 . 2 (𝐶 ∈ Cat → 𝐼 = ⟨(1st𝐼), (2nd𝐼)⟩)
65, 3eqeltrrd 2834 . . . . 5 (𝐶 ∈ Cat → ⟨(1st𝐼), (2nd𝐼)⟩ ∈ (𝐶 Func 𝐶))
7 df-br 5148 . . . . 5 ((1st𝐼)(𝐶 Func 𝐶)(2nd𝐼) ↔ ⟨(1st𝐼), (2nd𝐼)⟩ ∈ (𝐶 Func 𝐶))
86, 7sylibr 233 . . . 4 (𝐶 ∈ Cat → (1st𝐼)(𝐶 Func 𝐶)(2nd𝐼))
9 f1oi 6868 . . . . . 6 ( I ↾ (𝑥(Hom ‘𝐶)𝑦)):(𝑥(Hom ‘𝐶)𝑦)–1-1-onto→(𝑥(Hom ‘𝐶)𝑦)
10 eqid 2732 . . . . . . . 8 (Base‘𝐶) = (Base‘𝐶)
11 simpl 483 . . . . . . . 8 ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
12 eqid 2732 . . . . . . . 8 (Hom ‘𝐶) = (Hom ‘𝐶)
13 simprl 769 . . . . . . . 8 ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
14 simprr 771 . . . . . . . 8 ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
152, 10, 11, 12, 13, 14idfu2nd 17823 . . . . . . 7 ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝐼)𝑦) = ( I ↾ (𝑥(Hom ‘𝐶)𝑦)))
16 eqidd 2733 . . . . . . 7 ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐶)𝑦))
172, 10, 11, 13idfu1 17826 . . . . . . . 8 ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝐼)‘𝑥) = 𝑥)
182, 10, 11, 14idfu1 17826 . . . . . . . 8 ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝐼)‘𝑦) = 𝑦)
1917, 18oveq12d 7423 . . . . . . 7 ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (((1st𝐼)‘𝑥)(Hom ‘𝐶)((1st𝐼)‘𝑦)) = (𝑥(Hom ‘𝐶)𝑦))
2015, 16, 19f1oeq123d 6824 . . . . . 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 257 . . . . 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 3200 . . . 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 17863 . . . 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 583 . . 3 (𝐶 ∈ Cat → (1st𝐼)((𝐶 Full 𝐶) ∩ (𝐶 Faith 𝐶))(2nd𝐼))
25 df-br 5148 . . 3 ((1st𝐼)((𝐶 Full 𝐶) ∩ (𝐶 Faith 𝐶))(2nd𝐼) ↔ ⟨(1st𝐼), (2nd𝐼)⟩ ∈ ((𝐶 Full 𝐶) ∩ (𝐶 Faith 𝐶)))
2624, 25sylib 217 . 2 (𝐶 ∈ Cat → ⟨(1st𝐼), (2nd𝐼)⟩ ∈ ((𝐶 Full 𝐶) ∩ (𝐶 Faith 𝐶)))
275, 26eqeltrd 2833 1 (𝐶 ∈ Cat → 𝐼 ∈ ((𝐶 Full 𝐶) ∩ (𝐶 Faith 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3061  cin 3946  cop 4633   class class class wbr 5147   I cid 5572  cres 5677  Rel wrel 5680  1-1-ontowf1o 6539  cfv 6540  (class class class)co 7405  1st c1st 7969  2nd c2nd 7970  Basecbs 17140  Hom chom 17204  Catccat 17604   Func cfunc 17800  idfunccidfu 17801   Full cful 17849   Faith cfth 17850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-map 8818  df-ixp 8888  df-cat 17608  df-cid 17609  df-func 17804  df-idfu 17805  df-full 17851  df-fth 17852
This theorem is referenced by:  rescfth  17884
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