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Theorem idffth 16800
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 16729 . . 3 Rel (𝐶 Func 𝐶)
2 idffth.i . . . 4 𝐼 = (idfunc𝐶)
32idfucl 16748 . . 3 (𝐶 ∈ Cat → 𝐼 ∈ (𝐶 Func 𝐶))
4 1st2nd 7367 . . 3 ((Rel (𝐶 Func 𝐶) ∧ 𝐼 ∈ (𝐶 Func 𝐶)) → 𝐼 = ⟨(1st𝐼), (2nd𝐼)⟩)
51, 3, 4sylancr 575 . 2 (𝐶 ∈ Cat → 𝐼 = ⟨(1st𝐼), (2nd𝐼)⟩)
65, 3eqeltrrd 2851 . . . . 5 (𝐶 ∈ Cat → ⟨(1st𝐼), (2nd𝐼)⟩ ∈ (𝐶 Func 𝐶))
7 df-br 4788 . . . . 5 ((1st𝐼)(𝐶 Func 𝐶)(2nd𝐼) ↔ ⟨(1st𝐼), (2nd𝐼)⟩ ∈ (𝐶 Func 𝐶))
86, 7sylibr 224 . . . 4 (𝐶 ∈ Cat → (1st𝐼)(𝐶 Func 𝐶)(2nd𝐼))
9 f1oi 6316 . . . . . 6 ( I ↾ (𝑥(Hom ‘𝐶)𝑦)):(𝑥(Hom ‘𝐶)𝑦)–1-1-onto→(𝑥(Hom ‘𝐶)𝑦)
10 eqid 2771 . . . . . . . 8 (Base‘𝐶) = (Base‘𝐶)
11 simpl 468 . . . . . . . 8 ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
12 eqid 2771 . . . . . . . 8 (Hom ‘𝐶) = (Hom ‘𝐶)
13 simprl 754 . . . . . . . 8 ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
14 simprr 756 . . . . . . . 8 ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
152, 10, 11, 12, 13, 14idfu2nd 16744 . . . . . . 7 ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝐼)𝑦) = ( I ↾ (𝑥(Hom ‘𝐶)𝑦)))
16 eqidd 2772 . . . . . . 7 ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐶)𝑦))
172, 10, 11, 13idfu1 16747 . . . . . . . 8 ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝐼)‘𝑥) = 𝑥)
182, 10, 11, 14idfu1 16747 . . . . . . . 8 ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝐼)‘𝑦) = 𝑦)
1917, 18oveq12d 6814 . . . . . . 7 ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (((1st𝐼)‘𝑥)(Hom ‘𝐶)((1st𝐼)‘𝑦)) = (𝑥(Hom ‘𝐶)𝑦))
2015, 16, 19f1oeq123d 6275 . . . . . 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 3120 . . . 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 16783 . . . 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 572 . . 3 (𝐶 ∈ Cat → (1st𝐼)((𝐶 Full 𝐶) ∩ (𝐶 Faith 𝐶))(2nd𝐼))
25 df-br 4788 . . 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 2850 1 (𝐶 ∈ Cat → 𝐼 ∈ ((𝐶 Full 𝐶) ∩ (𝐶 Faith 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wral 3061  cin 3722  cop 4323   class class class wbr 4787   I cid 5157  cres 5252  Rel wrel 5255  1-1-ontowf1o 6029  cfv 6030  (class class class)co 6796  1st c1st 7317  2nd c2nd 7318  Basecbs 16064  Hom chom 16160  Catccat 16532   Func cfunc 16721  idfunccidfu 16722   Full cful 16769   Faith cfth 16770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-1st 7319  df-2nd 7320  df-map 8015  df-ixp 8067  df-cat 16536  df-cid 16537  df-func 16725  df-idfu 16726  df-full 16771  df-fth 16772
This theorem is referenced by:  rescfth  16804
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