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Theorem cofulid 17152
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 2798 . . . . . 6 (Base‘𝐷) = (Base‘𝐷)
3 cofulid.g . . . . . . . 8 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
4 funcrcl 17125 . . . . . . . 8 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
53, 4syl 17 . . . . . . 7 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
65simprd 499 . . . . . 6 (𝜑𝐷 ∈ Cat)
71, 2, 6idfu1st 17141 . . . . 5 (𝜑 → (1st𝐼) = ( I ↾ (Base‘𝐷)))
87coeq1d 5696 . . . 4 (𝜑 → ((1st𝐼) ∘ (1st𝐹)) = (( I ↾ (Base‘𝐷)) ∘ (1st𝐹)))
9 eqid 2798 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
10 relfunc 17124 . . . . . . 7 Rel (𝐶 Func 𝐷)
11 1st2ndbr 7723 . . . . . . 7 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
1210, 3, 11sylancr 590 . . . . . 6 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
139, 2, 12funcf1 17128 . . . . 5 (𝜑 → (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷))
14 fcoi2 6527 . . . . 5 ((1st𝐹):(Base‘𝐶)⟶(Base‘𝐷) → (( I ↾ (Base‘𝐷)) ∘ (1st𝐹)) = (1st𝐹))
1513, 14syl 17 . . . 4 (𝜑 → (( I ↾ (Base‘𝐷)) ∘ (1st𝐹)) = (1st𝐹))
168, 15eqtrd 2833 . . 3 (𝜑 → ((1st𝐼) ∘ (1st𝐹)) = (1st𝐹))
1763ad2ant1 1130 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
18 eqid 2798 . . . . . . . 8 (Hom ‘𝐷) = (Hom ‘𝐷)
1913ffvelrnda 6828 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
20193adant3 1129 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
2113ffvelrnda 6828 . . . . . . . . 9 ((𝜑𝑦 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑦) ∈ (Base‘𝐷))
22213adant2 1128 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑦) ∈ (Base‘𝐷))
231, 2, 17, 18, 20, 22idfu2nd 17139 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (((1st𝐹)‘𝑥)(2nd𝐼)((1st𝐹)‘𝑦)) = ( I ↾ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦))))
2423coeq1d 5696 . . . . . 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 2798 . . . . . . . 8 (Hom ‘𝐶) = (Hom ‘𝐶)
26123ad2ant1 1130 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
27 simp2 1134 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
28 simp3 1135 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑦 ∈ (Base‘𝐶))
299, 25, 18, 26, 27, 28funcf2 17130 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥(2nd𝐹)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
30 fcoi2 6527 . . . . . . 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 2833 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((((1st𝐹)‘𝑥)(2nd𝐼)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)) = (𝑥(2nd𝐹)𝑦))
3332mpoeq3dva 7210 . . . 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 17131 . . . . 5 (𝜑 → (2nd𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)))
35 fnov 7261 . . . . 5 ((2nd𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ (2nd𝐹) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd𝐹)𝑦)))
3634, 35sylib 221 . . . 4 (𝜑 → (2nd𝐹) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd𝐹)𝑦)))
3733, 36eqtr4d 2836 . . 3 (𝜑 → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st𝐹)‘𝑥)(2nd𝐼)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦))) = (2nd𝐹))
3816, 37opeq12d 4773 . 2 (𝜑 → ⟨((1st𝐼) ∘ (1st𝐹)), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st𝐹)‘𝑥)(2nd𝐼)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)))⟩ = ⟨(1st𝐹), (2nd𝐹)⟩)
391idfucl 17143 . . . 4 (𝐷 ∈ Cat → 𝐼 ∈ (𝐷 Func 𝐷))
406, 39syl 17 . . 3 (𝜑𝐼 ∈ (𝐷 Func 𝐷))
419, 3, 40cofuval 17144 . 2 (𝜑 → (𝐼func 𝐹) = ⟨((1st𝐼) ∘ (1st𝐹)), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st𝐹)‘𝑥)(2nd𝐼)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)))⟩)
42 1st2nd 7720 . . 3 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
4310, 3, 42sylancr 590 . 2 (𝜑𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
4438, 41, 433eqtr4d 2843 1 (𝜑 → (𝐼func 𝐹) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2111  cop 4531   class class class wbr 5030   I cid 5424   × cxp 5517  cres 5521  ccom 5523  Rel wrel 5524   Fn wfn 6319  wf 6320  cfv 6324  (class class class)co 7135  cmpo 7137  1st c1st 7669  2nd c2nd 7670  Basecbs 16475  Hom chom 16568  Catccat 16927   Func cfunc 17116  idfunccidfu 17117  func ccofu 17118
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-map 8391  df-ixp 8445  df-cat 16931  df-cid 16932  df-func 17120  df-idfu 17121  df-cofu 17122
This theorem is referenced by:  catccatid  17354
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