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Theorem cofulid 16373
 Description: The identity functor is a left identity for composition. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
cofulid.g (𝜑𝐹 ∈ (𝐶 Func 𝐷))
cofulid.1 𝐼 = (idfunc𝐷)
Assertion
Ref Expression
cofulid (𝜑 → (𝐼func 𝐹) = 𝐹)

Proof of Theorem cofulid
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cofulid.1 . . . . . 6 𝐼 = (idfunc𝐷)
2 eqid 2610 . . . . . 6 (Base‘𝐷) = (Base‘𝐷)
3 cofulid.g . . . . . . . 8 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
4 funcrcl 16346 . . . . . . . 8 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
53, 4syl 17 . . . . . . 7 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
65simprd 478 . . . . . 6 (𝜑𝐷 ∈ Cat)
71, 2, 6idfu1st 16362 . . . . 5 (𝜑 → (1st𝐼) = ( I ↾ (Base‘𝐷)))
87coeq1d 5205 . . . 4 (𝜑 → ((1st𝐼) ∘ (1st𝐹)) = (( I ↾ (Base‘𝐷)) ∘ (1st𝐹)))
9 eqid 2610 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
10 relfunc 16345 . . . . . . 7 Rel (𝐶 Func 𝐷)
11 1st2ndbr 7108 . . . . . . 7 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
1210, 3, 11sylancr 694 . . . . . 6 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
139, 2, 12funcf1 16349 . . . . 5 (𝜑 → (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷))
14 fcoi2 5992 . . . . 5 ((1st𝐹):(Base‘𝐶)⟶(Base‘𝐷) → (( I ↾ (Base‘𝐷)) ∘ (1st𝐹)) = (1st𝐹))
1513, 14syl 17 . . . 4 (𝜑 → (( I ↾ (Base‘𝐷)) ∘ (1st𝐹)) = (1st𝐹))
168, 15eqtrd 2644 . . 3 (𝜑 → ((1st𝐼) ∘ (1st𝐹)) = (1st𝐹))
1763ad2ant1 1075 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
18 eqid 2610 . . . . . . . 8 (Hom ‘𝐷) = (Hom ‘𝐷)
1913ffvelrnda 6267 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
20193adant3 1074 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
2113ffvelrnda 6267 . . . . . . . . 9 ((𝜑𝑦 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑦) ∈ (Base‘𝐷))
22213adant2 1073 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑦) ∈ (Base‘𝐷))
231, 2, 17, 18, 20, 22idfu2nd 16360 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (((1st𝐹)‘𝑥)(2nd𝐼)((1st𝐹)‘𝑦)) = ( I ↾ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦))))
2423coeq1d 5205 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((((1st𝐹)‘𝑥)(2nd𝐼)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)) = (( I ↾ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦))) ∘ (𝑥(2nd𝐹)𝑦)))
25 eqid 2610 . . . . . . . 8 (Hom ‘𝐶) = (Hom ‘𝐶)
26123ad2ant1 1075 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
27 simp2 1055 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
28 simp3 1056 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑦 ∈ (Base‘𝐶))
299, 25, 18, 26, 27, 28funcf2 16351 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥(2nd𝐹)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
30 fcoi2 5992 . . . . . . 7 ((𝑥(2nd𝐹)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)) → (( I ↾ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦))) ∘ (𝑥(2nd𝐹)𝑦)) = (𝑥(2nd𝐹)𝑦))
3129, 30syl 17 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (( I ↾ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦))) ∘ (𝑥(2nd𝐹)𝑦)) = (𝑥(2nd𝐹)𝑦))
3224, 31eqtrd 2644 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((((1st𝐹)‘𝑥)(2nd𝐼)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)) = (𝑥(2nd𝐹)𝑦))
3332mpt2eq3dva 6617 . . . 4 (𝜑 → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st𝐹)‘𝑥)(2nd𝐼)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦))) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd𝐹)𝑦)))
349, 12funcfn2 16352 . . . . 5 (𝜑 → (2nd𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)))
35 fnov 6666 . . . . 5 ((2nd𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ (2nd𝐹) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd𝐹)𝑦)))
3634, 35sylib 207 . . . 4 (𝜑 → (2nd𝐹) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd𝐹)𝑦)))
3733, 36eqtr4d 2647 . . 3 (𝜑 → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st𝐹)‘𝑥)(2nd𝐼)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦))) = (2nd𝐹))
3816, 37opeq12d 4348 . 2 (𝜑 → ⟨((1st𝐼) ∘ (1st𝐹)), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st𝐹)‘𝑥)(2nd𝐼)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)))⟩ = ⟨(1st𝐹), (2nd𝐹)⟩)
391idfucl 16364 . . . 4 (𝐷 ∈ Cat → 𝐼 ∈ (𝐷 Func 𝐷))
406, 39syl 17 . . 3 (𝜑𝐼 ∈ (𝐷 Func 𝐷))
419, 3, 40cofuval 16365 . 2 (𝜑 → (𝐼func 𝐹) = ⟨((1st𝐼) ∘ (1st𝐹)), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st𝐹)‘𝑥)(2nd𝐼)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)))⟩)
42 1st2nd 7105 . . 3 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
4310, 3, 42sylancr 694 . 2 (𝜑𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
4438, 41, 433eqtr4d 2654 1 (𝜑 → (𝐼func 𝐹) = 𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ⟨cop 4131   class class class wbr 4583   I cid 4948   × cxp 5036   ↾ cres 5040   ∘ ccom 5042  Rel wrel 5043   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1st c1st 7057  2nd c2nd 7058  Basecbs 15695  Hom chom 15779  Catccat 16148   Func cfunc 16337  idfunccidfu 16338   ∘func ccofu 16339 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-cofu 16343 This theorem is referenced by:  catccatid  16575
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