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Theorem idfucl 16742
Description: The identity functor is a functor. Example 3.20(1) of [Adamek] p. 30. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypothesis
Ref Expression
idfucl.i 𝐼 = (idfunc𝐶)
Assertion
Ref Expression
idfucl (𝐶 ∈ Cat → 𝐼 ∈ (𝐶 Func 𝐶))

Proof of Theorem idfucl
Dummy variables 𝑓 𝑔 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 idfucl.i . . . 4 𝐼 = (idfunc𝐶)
2 eqid 2760 . . . 4 (Base‘𝐶) = (Base‘𝐶)
3 id 22 . . . 4 (𝐶 ∈ Cat → 𝐶 ∈ Cat)
4 eqid 2760 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
51, 2, 3, 4idfuval 16737 . . 3 (𝐶 ∈ Cat → 𝐼 = ⟨( I ↾ (Base‘𝐶)), (𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧)))⟩)
65fveq2d 6356 . . . . 5 (𝐶 ∈ Cat → (2nd𝐼) = (2nd ‘⟨( I ↾ (Base‘𝐶)), (𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧)))⟩))
7 fvex 6362 . . . . . . 7 (Base‘𝐶) ∈ V
8 resiexg 7267 . . . . . . 7 ((Base‘𝐶) ∈ V → ( I ↾ (Base‘𝐶)) ∈ V)
97, 8ax-mp 5 . . . . . 6 ( I ↾ (Base‘𝐶)) ∈ V
107, 7xpex 7127 . . . . . . 7 ((Base‘𝐶) × (Base‘𝐶)) ∈ V
1110mptex 6650 . . . . . 6 (𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧))) ∈ V
129, 11op2nd 7342 . . . . 5 (2nd ‘⟨( I ↾ (Base‘𝐶)), (𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧)))⟩) = (𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧)))
136, 12syl6eq 2810 . . . 4 (𝐶 ∈ Cat → (2nd𝐼) = (𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧))))
1413opeq2d 4560 . . 3 (𝐶 ∈ Cat → ⟨( I ↾ (Base‘𝐶)), (2nd𝐼)⟩ = ⟨( I ↾ (Base‘𝐶)), (𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧)))⟩)
155, 14eqtr4d 2797 . 2 (𝐶 ∈ Cat → 𝐼 = ⟨( I ↾ (Base‘𝐶)), (2nd𝐼)⟩)
16 f1oi 6335 . . . . 5 ( I ↾ (Base‘𝐶)):(Base‘𝐶)–1-1-onto→(Base‘𝐶)
17 f1of 6298 . . . . 5 (( I ↾ (Base‘𝐶)):(Base‘𝐶)–1-1-onto→(Base‘𝐶) → ( I ↾ (Base‘𝐶)):(Base‘𝐶)⟶(Base‘𝐶))
1816, 17mp1i 13 . . . 4 (𝐶 ∈ Cat → ( I ↾ (Base‘𝐶)):(Base‘𝐶)⟶(Base‘𝐶))
19 f1oi 6335 . . . . . . . . . 10 ( I ↾ ((Hom ‘𝐶)‘𝑧)):((Hom ‘𝐶)‘𝑧)–1-1-onto→((Hom ‘𝐶)‘𝑧)
20 f1of 6298 . . . . . . . . . 10 (( I ↾ ((Hom ‘𝐶)‘𝑧)):((Hom ‘𝐶)‘𝑧)–1-1-onto→((Hom ‘𝐶)‘𝑧) → ( I ↾ ((Hom ‘𝐶)‘𝑧)):((Hom ‘𝐶)‘𝑧)⟶((Hom ‘𝐶)‘𝑧))
2119, 20ax-mp 5 . . . . . . . . 9 ( I ↾ ((Hom ‘𝐶)‘𝑧)):((Hom ‘𝐶)‘𝑧)⟶((Hom ‘𝐶)‘𝑧)
22 fvex 6362 . . . . . . . . . 10 ((Hom ‘𝐶)‘𝑧) ∈ V
2322, 22elmap 8052 . . . . . . . . 9 (( I ↾ ((Hom ‘𝐶)‘𝑧)) ∈ (((Hom ‘𝐶)‘𝑧) ↑𝑚 ((Hom ‘𝐶)‘𝑧)) ↔ ( I ↾ ((Hom ‘𝐶)‘𝑧)):((Hom ‘𝐶)‘𝑧)⟶((Hom ‘𝐶)‘𝑧))
2421, 23mpbir 221 . . . . . . . 8 ( I ↾ ((Hom ‘𝐶)‘𝑧)) ∈ (((Hom ‘𝐶)‘𝑧) ↑𝑚 ((Hom ‘𝐶)‘𝑧))
25 xp1st 7365 . . . . . . . . . . . . . 14 (𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶)) → (1st𝑧) ∈ (Base‘𝐶))
2625adantl 473 . . . . . . . . . . . . 13 ((𝐶 ∈ Cat ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) → (1st𝑧) ∈ (Base‘𝐶))
27 fvresi 6603 . . . . . . . . . . . . 13 ((1st𝑧) ∈ (Base‘𝐶) → (( I ↾ (Base‘𝐶))‘(1st𝑧)) = (1st𝑧))
2826, 27syl 17 . . . . . . . . . . . 12 ((𝐶 ∈ Cat ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) → (( I ↾ (Base‘𝐶))‘(1st𝑧)) = (1st𝑧))
29 xp2nd 7366 . . . . . . . . . . . . . 14 (𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶)) → (2nd𝑧) ∈ (Base‘𝐶))
3029adantl 473 . . . . . . . . . . . . 13 ((𝐶 ∈ Cat ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) → (2nd𝑧) ∈ (Base‘𝐶))
31 fvresi 6603 . . . . . . . . . . . . 13 ((2nd𝑧) ∈ (Base‘𝐶) → (( I ↾ (Base‘𝐶))‘(2nd𝑧)) = (2nd𝑧))
3230, 31syl 17 . . . . . . . . . . . 12 ((𝐶 ∈ Cat ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) → (( I ↾ (Base‘𝐶))‘(2nd𝑧)) = (2nd𝑧))
3328, 32oveq12d 6831 . . . . . . . . . . 11 ((𝐶 ∈ Cat ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((( I ↾ (Base‘𝐶))‘(1st𝑧))(Hom ‘𝐶)(( I ↾ (Base‘𝐶))‘(2nd𝑧))) = ((1st𝑧)(Hom ‘𝐶)(2nd𝑧)))
34 df-ov 6816 . . . . . . . . . . 11 ((1st𝑧)(Hom ‘𝐶)(2nd𝑧)) = ((Hom ‘𝐶)‘⟨(1st𝑧), (2nd𝑧)⟩)
3533, 34syl6eq 2810 . . . . . . . . . 10 ((𝐶 ∈ Cat ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((( I ↾ (Base‘𝐶))‘(1st𝑧))(Hom ‘𝐶)(( I ↾ (Base‘𝐶))‘(2nd𝑧))) = ((Hom ‘𝐶)‘⟨(1st𝑧), (2nd𝑧)⟩))
36 1st2nd2 7372 . . . . . . . . . . . 12 (𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶)) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
3736adantl 473 . . . . . . . . . . 11 ((𝐶 ∈ Cat ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
3837fveq2d 6356 . . . . . . . . . 10 ((𝐶 ∈ Cat ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Hom ‘𝐶)‘𝑧) = ((Hom ‘𝐶)‘⟨(1st𝑧), (2nd𝑧)⟩))
3935, 38eqtr4d 2797 . . . . . . . . 9 ((𝐶 ∈ Cat ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((( I ↾ (Base‘𝐶))‘(1st𝑧))(Hom ‘𝐶)(( I ↾ (Base‘𝐶))‘(2nd𝑧))) = ((Hom ‘𝐶)‘𝑧))
4039oveq1d 6828 . . . . . . . 8 ((𝐶 ∈ Cat ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) → (((( I ↾ (Base‘𝐶))‘(1st𝑧))(Hom ‘𝐶)(( I ↾ (Base‘𝐶))‘(2nd𝑧))) ↑𝑚 ((Hom ‘𝐶)‘𝑧)) = (((Hom ‘𝐶)‘𝑧) ↑𝑚 ((Hom ‘𝐶)‘𝑧)))
4124, 40syl5eleqr 2846 . . . . . . 7 ((𝐶 ∈ Cat ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ( I ↾ ((Hom ‘𝐶)‘𝑧)) ∈ (((( I ↾ (Base‘𝐶))‘(1st𝑧))(Hom ‘𝐶)(( I ↾ (Base‘𝐶))‘(2nd𝑧))) ↑𝑚 ((Hom ‘𝐶)‘𝑧)))
4241ralrimiva 3104 . . . . . 6 (𝐶 ∈ Cat → ∀𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))( I ↾ ((Hom ‘𝐶)‘𝑧)) ∈ (((( I ↾ (Base‘𝐶))‘(1st𝑧))(Hom ‘𝐶)(( I ↾ (Base‘𝐶))‘(2nd𝑧))) ↑𝑚 ((Hom ‘𝐶)‘𝑧)))
43 mptelixpg 8111 . . . . . . 7 (((Base‘𝐶) × (Base‘𝐶)) ∈ V → ((𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧))) ∈ X𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))(((( I ↾ (Base‘𝐶))‘(1st𝑧))(Hom ‘𝐶)(( I ↾ (Base‘𝐶))‘(2nd𝑧))) ↑𝑚 ((Hom ‘𝐶)‘𝑧)) ↔ ∀𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))( I ↾ ((Hom ‘𝐶)‘𝑧)) ∈ (((( I ↾ (Base‘𝐶))‘(1st𝑧))(Hom ‘𝐶)(( I ↾ (Base‘𝐶))‘(2nd𝑧))) ↑𝑚 ((Hom ‘𝐶)‘𝑧))))
4410, 43ax-mp 5 . . . . . 6 ((𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧))) ∈ X𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))(((( I ↾ (Base‘𝐶))‘(1st𝑧))(Hom ‘𝐶)(( I ↾ (Base‘𝐶))‘(2nd𝑧))) ↑𝑚 ((Hom ‘𝐶)‘𝑧)) ↔ ∀𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))( I ↾ ((Hom ‘𝐶)‘𝑧)) ∈ (((( I ↾ (Base‘𝐶))‘(1st𝑧))(Hom ‘𝐶)(( I ↾ (Base‘𝐶))‘(2nd𝑧))) ↑𝑚 ((Hom ‘𝐶)‘𝑧)))
4542, 44sylibr 224 . . . . 5 (𝐶 ∈ Cat → (𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧))) ∈ X𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))(((( I ↾ (Base‘𝐶))‘(1st𝑧))(Hom ‘𝐶)(( I ↾ (Base‘𝐶))‘(2nd𝑧))) ↑𝑚 ((Hom ‘𝐶)‘𝑧)))
4613, 45eqeltrd 2839 . . . 4 (𝐶 ∈ Cat → (2nd𝐼) ∈ X𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))(((( I ↾ (Base‘𝐶))‘(1st𝑧))(Hom ‘𝐶)(( I ↾ (Base‘𝐶))‘(2nd𝑧))) ↑𝑚 ((Hom ‘𝐶)‘𝑧)))
47 eqid 2760 . . . . . . . . 9 (Id‘𝐶) = (Id‘𝐶)
48 simpl 474 . . . . . . . . 9 ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐶 ∈ Cat)
49 simpr 479 . . . . . . . . 9 ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
502, 4, 47, 48, 49catidcl 16544 . . . . . . . 8 ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥))
51 fvresi 6603 . . . . . . . 8 (((Id‘𝐶)‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥) → (( I ↾ (𝑥(Hom ‘𝐶)𝑥))‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐶)‘𝑥))
5250, 51syl 17 . . . . . . 7 ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → (( I ↾ (𝑥(Hom ‘𝐶)𝑥))‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐶)‘𝑥))
531, 2, 48, 4, 49, 49idfu2nd 16738 . . . . . . . 8 ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑥(2nd𝐼)𝑥) = ( I ↾ (𝑥(Hom ‘𝐶)𝑥)))
5453fveq1d 6354 . . . . . . 7 ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝑥(2nd𝐼)𝑥)‘((Id‘𝐶)‘𝑥)) = (( I ↾ (𝑥(Hom ‘𝐶)𝑥))‘((Id‘𝐶)‘𝑥)))
55 fvresi 6603 . . . . . . . . 9 (𝑥 ∈ (Base‘𝐶) → (( I ↾ (Base‘𝐶))‘𝑥) = 𝑥)
5655adantl 473 . . . . . . . 8 ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → (( I ↾ (Base‘𝐶))‘𝑥) = 𝑥)
5756fveq2d 6356 . . . . . . 7 ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘(( I ↾ (Base‘𝐶))‘𝑥)) = ((Id‘𝐶)‘𝑥))
5852, 54, 573eqtr4d 2804 . . . . . 6 ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝑥(2nd𝐼)𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐶)‘(( I ↾ (Base‘𝐶))‘𝑥)))
59 eqid 2760 . . . . . . . . . . 11 (comp‘𝐶) = (comp‘𝐶)
6048ad2antrr 764 . . . . . . . . . . 11 ((((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝐶 ∈ Cat)
6149ad2antrr 764 . . . . . . . . . . 11 ((((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑥 ∈ (Base‘𝐶))
62 simplrl 819 . . . . . . . . . . 11 ((((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑦 ∈ (Base‘𝐶))
63 simplrr 820 . . . . . . . . . . 11 ((((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑧 ∈ (Base‘𝐶))
64 simprl 811 . . . . . . . . . . 11 ((((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
65 simprr 813 . . . . . . . . . . 11 ((((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))
662, 4, 59, 60, 61, 62, 63, 64, 65catcocl 16547 . . . . . . . . . 10 ((((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧))
67 fvresi 6603 . . . . . . . . . 10 ((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) → (( I ↾ (𝑥(Hom ‘𝐶)𝑧))‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))
6866, 67syl 17 . . . . . . . . 9 ((((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (( I ↾ (𝑥(Hom ‘𝐶)𝑧))‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))
691, 2, 60, 4, 61, 63idfu2nd 16738 . . . . . . . . . 10 ((((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑥(2nd𝐼)𝑧) = ( I ↾ (𝑥(Hom ‘𝐶)𝑧)))
7069fveq1d 6354 . . . . . . . . 9 ((((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑥(2nd𝐼)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (( I ↾ (𝑥(Hom ‘𝐶)𝑧))‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))
7161, 55syl 17 . . . . . . . . . . . 12 ((((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (( I ↾ (Base‘𝐶))‘𝑥) = 𝑥)
72 fvresi 6603 . . . . . . . . . . . . 13 (𝑦 ∈ (Base‘𝐶) → (( I ↾ (Base‘𝐶))‘𝑦) = 𝑦)
7362, 72syl 17 . . . . . . . . . . . 12 ((((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (( I ↾ (Base‘𝐶))‘𝑦) = 𝑦)
7471, 73opeq12d 4561 . . . . . . . . . . 11 ((((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ⟨(( I ↾ (Base‘𝐶))‘𝑥), (( I ↾ (Base‘𝐶))‘𝑦)⟩ = ⟨𝑥, 𝑦⟩)
75 fvresi 6603 . . . . . . . . . . . 12 (𝑧 ∈ (Base‘𝐶) → (( I ↾ (Base‘𝐶))‘𝑧) = 𝑧)
7663, 75syl 17 . . . . . . . . . . 11 ((((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (( I ↾ (Base‘𝐶))‘𝑧) = 𝑧)
7774, 76oveq12d 6831 . . . . . . . . . 10 ((((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (⟨(( I ↾ (Base‘𝐶))‘𝑥), (( I ↾ (Base‘𝐶))‘𝑦)⟩(comp‘𝐶)(( I ↾ (Base‘𝐶))‘𝑧)) = (⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧))
781, 2, 60, 4, 62, 63, 65idfu2 16739 . . . . . . . . . 10 ((((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑦(2nd𝐼)𝑧)‘𝑔) = 𝑔)
791, 2, 60, 4, 61, 62, 64idfu2 16739 . . . . . . . . . 10 ((((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑥(2nd𝐼)𝑦)‘𝑓) = 𝑓)
8077, 78, 79oveq123d 6834 . . . . . . . . 9 ((((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (((𝑦(2nd𝐼)𝑧)‘𝑔)(⟨(( I ↾ (Base‘𝐶))‘𝑥), (( I ↾ (Base‘𝐶))‘𝑦)⟩(comp‘𝐶)(( I ↾ (Base‘𝐶))‘𝑧))((𝑥(2nd𝐼)𝑦)‘𝑓)) = (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))
8168, 70, 803eqtr4d 2804 . . . . . . . 8 ((((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑥(2nd𝐼)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦(2nd𝐼)𝑧)‘𝑔)(⟨(( I ↾ (Base‘𝐶))‘𝑥), (( I ↾ (Base‘𝐶))‘𝑦)⟩(comp‘𝐶)(( I ↾ (Base‘𝐶))‘𝑧))((𝑥(2nd𝐼)𝑦)‘𝑓)))
8281ralrimivva 3109 . . . . . . 7 (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑥(2nd𝐼)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦(2nd𝐼)𝑧)‘𝑔)(⟨(( I ↾ (Base‘𝐶))‘𝑥), (( I ↾ (Base‘𝐶))‘𝑦)⟩(comp‘𝐶)(( I ↾ (Base‘𝐶))‘𝑧))((𝑥(2nd𝐼)𝑦)‘𝑓)))
8382ralrimivva 3109 . . . . . 6 ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑥(2nd𝐼)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦(2nd𝐼)𝑧)‘𝑔)(⟨(( I ↾ (Base‘𝐶))‘𝑥), (( I ↾ (Base‘𝐶))‘𝑦)⟩(comp‘𝐶)(( I ↾ (Base‘𝐶))‘𝑧))((𝑥(2nd𝐼)𝑦)‘𝑓)))
8458, 83jca 555 . . . . 5 ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → (((𝑥(2nd𝐼)𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐶)‘(( I ↾ (Base‘𝐶))‘𝑥)) ∧ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑥(2nd𝐼)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦(2nd𝐼)𝑧)‘𝑔)(⟨(( I ↾ (Base‘𝐶))‘𝑥), (( I ↾ (Base‘𝐶))‘𝑦)⟩(comp‘𝐶)(( I ↾ (Base‘𝐶))‘𝑧))((𝑥(2nd𝐼)𝑦)‘𝑓))))
8584ralrimiva 3104 . . . 4 (𝐶 ∈ Cat → ∀𝑥 ∈ (Base‘𝐶)(((𝑥(2nd𝐼)𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐶)‘(( I ↾ (Base‘𝐶))‘𝑥)) ∧ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑥(2nd𝐼)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦(2nd𝐼)𝑧)‘𝑔)(⟨(( I ↾ (Base‘𝐶))‘𝑥), (( I ↾ (Base‘𝐶))‘𝑦)⟩(comp‘𝐶)(( I ↾ (Base‘𝐶))‘𝑧))((𝑥(2nd𝐼)𝑦)‘𝑓))))
862, 2, 4, 4, 47, 47, 59, 59, 3, 3isfunc 16725 . . . 4 (𝐶 ∈ Cat → (( I ↾ (Base‘𝐶))(𝐶 Func 𝐶)(2nd𝐼) ↔ (( I ↾ (Base‘𝐶)):(Base‘𝐶)⟶(Base‘𝐶) ∧ (2nd𝐼) ∈ X𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))(((( I ↾ (Base‘𝐶))‘(1st𝑧))(Hom ‘𝐶)(( I ↾ (Base‘𝐶))‘(2nd𝑧))) ↑𝑚 ((Hom ‘𝐶)‘𝑧)) ∧ ∀𝑥 ∈ (Base‘𝐶)(((𝑥(2nd𝐼)𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐶)‘(( I ↾ (Base‘𝐶))‘𝑥)) ∧ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑥(2nd𝐼)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦(2nd𝐼)𝑧)‘𝑔)(⟨(( I ↾ (Base‘𝐶))‘𝑥), (( I ↾ (Base‘𝐶))‘𝑦)⟩(comp‘𝐶)(( I ↾ (Base‘𝐶))‘𝑧))((𝑥(2nd𝐼)𝑦)‘𝑓))))))
8718, 46, 85, 86mpbir3and 1428 . . 3 (𝐶 ∈ Cat → ( I ↾ (Base‘𝐶))(𝐶 Func 𝐶)(2nd𝐼))
88 df-br 4805 . . 3 (( I ↾ (Base‘𝐶))(𝐶 Func 𝐶)(2nd𝐼) ↔ ⟨( I ↾ (Base‘𝐶)), (2nd𝐼)⟩ ∈ (𝐶 Func 𝐶))
8987, 88sylib 208 . 2 (𝐶 ∈ Cat → ⟨( I ↾ (Base‘𝐶)), (2nd𝐼)⟩ ∈ (𝐶 Func 𝐶))
9015, 89eqeltrd 2839 1 (𝐶 ∈ Cat → 𝐼 ∈ (𝐶 Func 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050  Vcvv 3340  cop 4327   class class class wbr 4804  cmpt 4881   I cid 5173   × cxp 5264  cres 5268  wf 6045  1-1-ontowf1o 6048  cfv 6049  (class class class)co 6813  1st c1st 7331  2nd c2nd 7332  𝑚 cmap 8023  Xcixp 8074  Basecbs 16059  Hom chom 16154  compcco 16155  Catccat 16526  Idccid 16527   Func cfunc 16715  idfunccidfu 16716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-map 8025  df-ixp 8075  df-cat 16530  df-cid 16531  df-func 16719  df-idfu 16720
This theorem is referenced by:  cofulid  16751  cofurid  16752  idffth  16794  ressffth  16799  catccatid  16953  curf2ndf  17088
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