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Theorem cofurid 17847
Description: The identity functor is a right identity for composition. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
cofulid.g (𝜑𝐹 ∈ (𝐶 Func 𝐷))
cofurid.1 𝐼 = (idfunc𝐶)
Assertion
Ref Expression
cofurid (𝜑 → (𝐹func 𝐼) = 𝐹)

Proof of Theorem cofurid
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cofurid.1 . . . . . 6 𝐼 = (idfunc𝐶)
2 eqid 2726 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
3 cofulid.g . . . . . . . 8 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
4 funcrcl 17819 . . . . . . . 8 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
53, 4syl 17 . . . . . . 7 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
65simpld 494 . . . . . 6 (𝜑𝐶 ∈ Cat)
71, 2, 6idfu1st 17835 . . . . 5 (𝜑 → (1st𝐼) = ( I ↾ (Base‘𝐶)))
87coeq2d 5855 . . . 4 (𝜑 → ((1st𝐹) ∘ (1st𝐼)) = ((1st𝐹) ∘ ( I ↾ (Base‘𝐶))))
9 eqid 2726 . . . . . 6 (Base‘𝐷) = (Base‘𝐷)
10 relfunc 17818 . . . . . . 7 Rel (𝐶 Func 𝐷)
11 1st2ndbr 8024 . . . . . . 7 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
1210, 3, 11sylancr 586 . . . . . 6 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
132, 9, 12funcf1 17822 . . . . 5 (𝜑 → (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷))
14 fcoi1 6758 . . . . 5 ((1st𝐹):(Base‘𝐶)⟶(Base‘𝐷) → ((1st𝐹) ∘ ( I ↾ (Base‘𝐶))) = (1st𝐹))
1513, 14syl 17 . . . 4 (𝜑 → ((1st𝐹) ∘ ( I ↾ (Base‘𝐶))) = (1st𝐹))
168, 15eqtrd 2766 . . 3 (𝜑 → ((1st𝐹) ∘ (1st𝐼)) = (1st𝐹))
1773ad2ant1 1130 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (1st𝐼) = ( I ↾ (Base‘𝐶)))
1817fveq1d 6886 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st𝐼)‘𝑥) = (( I ↾ (Base‘𝐶))‘𝑥))
19 fvresi 7166 . . . . . . . . . 10 (𝑥 ∈ (Base‘𝐶) → (( I ↾ (Base‘𝐶))‘𝑥) = 𝑥)
20193ad2ant2 1131 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (( I ↾ (Base‘𝐶))‘𝑥) = 𝑥)
2118, 20eqtrd 2766 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st𝐼)‘𝑥) = 𝑥)
2217fveq1d 6886 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st𝐼)‘𝑦) = (( I ↾ (Base‘𝐶))‘𝑦))
23 fvresi 7166 . . . . . . . . . 10 (𝑦 ∈ (Base‘𝐶) → (( I ↾ (Base‘𝐶))‘𝑦) = 𝑦)
24233ad2ant3 1132 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (( I ↾ (Base‘𝐶))‘𝑦) = 𝑦)
2522, 24eqtrd 2766 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st𝐼)‘𝑦) = 𝑦)
2621, 25oveq12d 7422 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (((1st𝐼)‘𝑥)(2nd𝐹)((1st𝐼)‘𝑦)) = (𝑥(2nd𝐹)𝑦))
2763ad2ant1 1130 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝐶 ∈ Cat)
28 eqid 2726 . . . . . . . 8 (Hom ‘𝐶) = (Hom ‘𝐶)
29 simp2 1134 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
30 simp3 1135 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑦 ∈ (Base‘𝐶))
311, 2, 27, 28, 29, 30idfu2nd 17833 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥(2nd𝐼)𝑦) = ( I ↾ (𝑥(Hom ‘𝐶)𝑦)))
3226, 31coeq12d 5857 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((((1st𝐼)‘𝑥)(2nd𝐹)((1st𝐼)‘𝑦)) ∘ (𝑥(2nd𝐼)𝑦)) = ((𝑥(2nd𝐹)𝑦) ∘ ( I ↾ (𝑥(Hom ‘𝐶)𝑦))))
33 eqid 2726 . . . . . . . 8 (Hom ‘𝐷) = (Hom ‘𝐷)
34123ad2ant1 1130 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
352, 28, 33, 34, 29, 30funcf2 17824 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥(2nd𝐹)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
36 fcoi1 6758 . . . . . . 7 ((𝑥(2nd𝐹)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)) → ((𝑥(2nd𝐹)𝑦) ∘ ( I ↾ (𝑥(Hom ‘𝐶)𝑦))) = (𝑥(2nd𝐹)𝑦))
3735, 36syl 17 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((𝑥(2nd𝐹)𝑦) ∘ ( I ↾ (𝑥(Hom ‘𝐶)𝑦))) = (𝑥(2nd𝐹)𝑦))
3832, 37eqtrd 2766 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((((1st𝐼)‘𝑥)(2nd𝐹)((1st𝐼)‘𝑦)) ∘ (𝑥(2nd𝐼)𝑦)) = (𝑥(2nd𝐹)𝑦))
3938mpoeq3dva 7481 . . . 4 (𝜑 → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st𝐼)‘𝑥)(2nd𝐹)((1st𝐼)‘𝑦)) ∘ (𝑥(2nd𝐼)𝑦))) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd𝐹)𝑦)))
402, 12funcfn2 17825 . . . . 5 (𝜑 → (2nd𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)))
41 fnov 7535 . . . . 5 ((2nd𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ (2nd𝐹) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd𝐹)𝑦)))
4240, 41sylib 217 . . . 4 (𝜑 → (2nd𝐹) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd𝐹)𝑦)))
4339, 42eqtr4d 2769 . . 3 (𝜑 → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st𝐼)‘𝑥)(2nd𝐹)((1st𝐼)‘𝑦)) ∘ (𝑥(2nd𝐼)𝑦))) = (2nd𝐹))
4416, 43opeq12d 4876 . 2 (𝜑 → ⟨((1st𝐹) ∘ (1st𝐼)), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st𝐼)‘𝑥)(2nd𝐹)((1st𝐼)‘𝑦)) ∘ (𝑥(2nd𝐼)𝑦)))⟩ = ⟨(1st𝐹), (2nd𝐹)⟩)
451idfucl 17837 . . . 4 (𝐶 ∈ Cat → 𝐼 ∈ (𝐶 Func 𝐶))
466, 45syl 17 . . 3 (𝜑𝐼 ∈ (𝐶 Func 𝐶))
472, 46, 3cofuval 17838 . 2 (𝜑 → (𝐹func 𝐼) = ⟨((1st𝐹) ∘ (1st𝐼)), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st𝐼)‘𝑥)(2nd𝐹)((1st𝐼)‘𝑦)) ∘ (𝑥(2nd𝐼)𝑦)))⟩)
48 1st2nd 8021 . . 3 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
4910, 3, 48sylancr 586 . 2 (𝜑𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
5044, 47, 493eqtr4d 2776 1 (𝜑 → (𝐹func 𝐼) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084   = wceq 1533  wcel 2098  cop 4629   class class class wbr 5141   I cid 5566   × cxp 5667  cres 5671  ccom 5673  Rel wrel 5674   Fn wfn 6531  wf 6532  cfv 6536  (class class class)co 7404  cmpo 7406  1st c1st 7969  2nd c2nd 7970  Basecbs 17150  Hom chom 17214  Catccat 17614   Func cfunc 17810  idfunccidfu 17811  func ccofu 17812
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7971  df-2nd 7972  df-map 8821  df-ixp 8891  df-cat 17618  df-cid 17619  df-func 17814  df-idfu 17815  df-cofu 17816
This theorem is referenced by:  catccatid  18065
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