MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cofurid Structured version   Visualization version   GIF version

Theorem cofurid 16818
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 2765 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
3 cofulid.g . . . . . . . 8 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
4 funcrcl 16790 . . . . . . . 8 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
53, 4syl 17 . . . . . . 7 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
65simpld 488 . . . . . 6 (𝜑𝐶 ∈ Cat)
71, 2, 6idfu1st 16806 . . . . 5 (𝜑 → (1st𝐼) = ( I ↾ (Base‘𝐶)))
87coeq2d 5453 . . . 4 (𝜑 → ((1st𝐹) ∘ (1st𝐼)) = ((1st𝐹) ∘ ( I ↾ (Base‘𝐶))))
9 eqid 2765 . . . . . 6 (Base‘𝐷) = (Base‘𝐷)
10 relfunc 16789 . . . . . . 7 Rel (𝐶 Func 𝐷)
11 1st2ndbr 7417 . . . . . . 7 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
1210, 3, 11sylancr 581 . . . . . 6 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
132, 9, 12funcf1 16793 . . . . 5 (𝜑 → (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷))
14 fcoi1 6260 . . . . 5 ((1st𝐹):(Base‘𝐶)⟶(Base‘𝐷) → ((1st𝐹) ∘ ( I ↾ (Base‘𝐶))) = (1st𝐹))
1513, 14syl 17 . . . 4 (𝜑 → ((1st𝐹) ∘ ( I ↾ (Base‘𝐶))) = (1st𝐹))
168, 15eqtrd 2799 . . 3 (𝜑 → ((1st𝐹) ∘ (1st𝐼)) = (1st𝐹))
1773ad2ant1 1163 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (1st𝐼) = ( I ↾ (Base‘𝐶)))
1817fveq1d 6377 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st𝐼)‘𝑥) = (( I ↾ (Base‘𝐶))‘𝑥))
19 fvresi 6632 . . . . . . . . . 10 (𝑥 ∈ (Base‘𝐶) → (( I ↾ (Base‘𝐶))‘𝑥) = 𝑥)
20193ad2ant2 1164 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (( I ↾ (Base‘𝐶))‘𝑥) = 𝑥)
2118, 20eqtrd 2799 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st𝐼)‘𝑥) = 𝑥)
2217fveq1d 6377 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st𝐼)‘𝑦) = (( I ↾ (Base‘𝐶))‘𝑦))
23 fvresi 6632 . . . . . . . . . 10 (𝑦 ∈ (Base‘𝐶) → (( I ↾ (Base‘𝐶))‘𝑦) = 𝑦)
24233ad2ant3 1165 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (( I ↾ (Base‘𝐶))‘𝑦) = 𝑦)
2522, 24eqtrd 2799 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st𝐼)‘𝑦) = 𝑦)
2621, 25oveq12d 6860 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (((1st𝐼)‘𝑥)(2nd𝐹)((1st𝐼)‘𝑦)) = (𝑥(2nd𝐹)𝑦))
2763ad2ant1 1163 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝐶 ∈ Cat)
28 eqid 2765 . . . . . . . 8 (Hom ‘𝐶) = (Hom ‘𝐶)
29 simp2 1167 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
30 simp3 1168 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑦 ∈ (Base‘𝐶))
311, 2, 27, 28, 29, 30idfu2nd 16804 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥(2nd𝐼)𝑦) = ( I ↾ (𝑥(Hom ‘𝐶)𝑦)))
3226, 31coeq12d 5455 . . . . . 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 2765 . . . . . . . 8 (Hom ‘𝐷) = (Hom ‘𝐷)
34123ad2ant1 1163 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
352, 28, 33, 34, 29, 30funcf2 16795 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥(2nd𝐹)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
36 fcoi1 6260 . . . . . . 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 2799 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((((1st𝐼)‘𝑥)(2nd𝐹)((1st𝐼)‘𝑦)) ∘ (𝑥(2nd𝐼)𝑦)) = (𝑥(2nd𝐹)𝑦))
3938mpt2eq3dva 6917 . . . 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 16796 . . . . 5 (𝜑 → (2nd𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)))
41 fnov 6966 . . . . 5 ((2nd𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ (2nd𝐹) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd𝐹)𝑦)))
4240, 41sylib 209 . . . 4 (𝜑 → (2nd𝐹) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd𝐹)𝑦)))
4339, 42eqtr4d 2802 . . 3 (𝜑 → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st𝐼)‘𝑥)(2nd𝐹)((1st𝐼)‘𝑦)) ∘ (𝑥(2nd𝐼)𝑦))) = (2nd𝐹))
4416, 43opeq12d 4567 . 2 (𝜑 → ⟨((1st𝐹) ∘ (1st𝐼)), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st𝐼)‘𝑥)(2nd𝐹)((1st𝐼)‘𝑦)) ∘ (𝑥(2nd𝐼)𝑦)))⟩ = ⟨(1st𝐹), (2nd𝐹)⟩)
451idfucl 16808 . . . 4 (𝐶 ∈ Cat → 𝐼 ∈ (𝐶 Func 𝐶))
466, 45syl 17 . . 3 (𝜑𝐼 ∈ (𝐶 Func 𝐶))
472, 46, 3cofuval 16809 . 2 (𝜑 → (𝐹func 𝐼) = ⟨((1st𝐹) ∘ (1st𝐼)), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st𝐼)‘𝑥)(2nd𝐹)((1st𝐼)‘𝑦)) ∘ (𝑥(2nd𝐼)𝑦)))⟩)
48 1st2nd 7414 . . 3 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
4910, 3, 48sylancr 581 . 2 (𝜑𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
5044, 47, 493eqtr4d 2809 1 (𝜑 → (𝐹func 𝐼) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1107   = wceq 1652  wcel 2155  cop 4340   class class class wbr 4809   I cid 5184   × cxp 5275  cres 5279  ccom 5281  Rel wrel 5282   Fn wfn 6063  wf 6064  cfv 6068  (class class class)co 6842  cmpt2 6844  1st c1st 7364  2nd c2nd 7365  Basecbs 16132  Hom chom 16227  Catccat 16592   Func cfunc 16781  idfunccidfu 16782  func ccofu 16783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-1st 7366  df-2nd 7367  df-map 8062  df-ixp 8114  df-cat 16596  df-cid 16597  df-func 16785  df-idfu 16786  df-cofu 16787
This theorem is referenced by:  catccatid  17019
  Copyright terms: Public domain W3C validator