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Theorem cofulid 17844
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 2732 . . . . . 6 (Base‘𝐷) = (Base‘𝐷)
3 cofulid.g . . . . . . . 8 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
4 funcrcl 17817 . . . . . . . 8 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
53, 4syl 17 . . . . . . 7 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
65simprd 496 . . . . . 6 (𝜑𝐷 ∈ Cat)
71, 2, 6idfu1st 17833 . . . . 5 (𝜑 → (1st𝐼) = ( I ↾ (Base‘𝐷)))
87coeq1d 5861 . . . 4 (𝜑 → ((1st𝐼) ∘ (1st𝐹)) = (( I ↾ (Base‘𝐷)) ∘ (1st𝐹)))
9 eqid 2732 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
10 relfunc 17816 . . . . . . 7 Rel (𝐶 Func 𝐷)
11 1st2ndbr 8030 . . . . . . 7 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
1210, 3, 11sylancr 587 . . . . . 6 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
139, 2, 12funcf1 17820 . . . . 5 (𝜑 → (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷))
14 fcoi2 6766 . . . . 5 ((1st𝐹):(Base‘𝐶)⟶(Base‘𝐷) → (( I ↾ (Base‘𝐷)) ∘ (1st𝐹)) = (1st𝐹))
1513, 14syl 17 . . . 4 (𝜑 → (( I ↾ (Base‘𝐷)) ∘ (1st𝐹)) = (1st𝐹))
168, 15eqtrd 2772 . . 3 (𝜑 → ((1st𝐼) ∘ (1st𝐹)) = (1st𝐹))
1763ad2ant1 1133 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
18 eqid 2732 . . . . . . . 8 (Hom ‘𝐷) = (Hom ‘𝐷)
1913ffvelcdmda 7086 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
20193adant3 1132 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
2113ffvelcdmda 7086 . . . . . . . . 9 ((𝜑𝑦 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑦) ∈ (Base‘𝐷))
22213adant2 1131 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑦) ∈ (Base‘𝐷))
231, 2, 17, 18, 20, 22idfu2nd 17831 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (((1st𝐹)‘𝑥)(2nd𝐼)((1st𝐹)‘𝑦)) = ( I ↾ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦))))
2423coeq1d 5861 . . . . . 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 2732 . . . . . . . 8 (Hom ‘𝐶) = (Hom ‘𝐶)
26123ad2ant1 1133 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
27 simp2 1137 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
28 simp3 1138 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑦 ∈ (Base‘𝐶))
299, 25, 18, 26, 27, 28funcf2 17822 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥(2nd𝐹)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
30 fcoi2 6766 . . . . . . 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 2772 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((((1st𝐹)‘𝑥)(2nd𝐼)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)) = (𝑥(2nd𝐹)𝑦))
3332mpoeq3dva 7488 . . . 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 17823 . . . . 5 (𝜑 → (2nd𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)))
35 fnov 7542 . . . . 5 ((2nd𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ (2nd𝐹) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd𝐹)𝑦)))
3634, 35sylib 217 . . . 4 (𝜑 → (2nd𝐹) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd𝐹)𝑦)))
3733, 36eqtr4d 2775 . . 3 (𝜑 → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st𝐹)‘𝑥)(2nd𝐼)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦))) = (2nd𝐹))
3816, 37opeq12d 4881 . 2 (𝜑 → ⟨((1st𝐼) ∘ (1st𝐹)), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st𝐹)‘𝑥)(2nd𝐼)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)))⟩ = ⟨(1st𝐹), (2nd𝐹)⟩)
391idfucl 17835 . . . 4 (𝐷 ∈ Cat → 𝐼 ∈ (𝐷 Func 𝐷))
406, 39syl 17 . . 3 (𝜑𝐼 ∈ (𝐷 Func 𝐷))
419, 3, 40cofuval 17836 . 2 (𝜑 → (𝐼func 𝐹) = ⟨((1st𝐼) ∘ (1st𝐹)), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st𝐹)‘𝑥)(2nd𝐼)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)))⟩)
42 1st2nd 8027 . . 3 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
4310, 3, 42sylancr 587 . 2 (𝜑𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
4438, 41, 433eqtr4d 2782 1 (𝜑 → (𝐼func 𝐹) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  cop 4634   class class class wbr 5148   I cid 5573   × cxp 5674  cres 5678  ccom 5680  Rel wrel 5681   Fn wfn 6538  wf 6539  cfv 6543  (class class class)co 7411  cmpo 7413  1st c1st 7975  2nd c2nd 7976  Basecbs 17148  Hom chom 17212  Catccat 17612   Func cfunc 17808  idfunccidfu 17809  func ccofu 17810
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 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
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 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-map 8824  df-ixp 8894  df-cat 17616  df-cid 17617  df-func 17812  df-idfu 17813  df-cofu 17814
This theorem is referenced by:  catccatid  18060
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