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Theorem idfucl 17141
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 2826 . . . 4 (Base‘𝐶) = (Base‘𝐶)
3 id 22 . . . 4 (𝐶 ∈ Cat → 𝐶 ∈ Cat)
4 eqid 2826 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
51, 2, 3, 4idfuval 17136 . . 3 (𝐶 ∈ Cat → 𝐼 = ⟨( I ↾ (Base‘𝐶)), (𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧)))⟩)
65fveq2d 6671 . . . . 5 (𝐶 ∈ Cat → (2nd𝐼) = (2nd ‘⟨( I ↾ (Base‘𝐶)), (𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧)))⟩))
7 fvex 6680 . . . . . . 7 (Base‘𝐶) ∈ V
8 resiexg 7607 . . . . . . 7 ((Base‘𝐶) ∈ V → ( I ↾ (Base‘𝐶)) ∈ V)
97, 8ax-mp 5 . . . . . 6 ( I ↾ (Base‘𝐶)) ∈ V
107, 7xpex 7465 . . . . . . 7 ((Base‘𝐶) × (Base‘𝐶)) ∈ V
1110mptex 6981 . . . . . 6 (𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧))) ∈ V
129, 11op2nd 7689 . . . . 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 2877 . . . 4 (𝐶 ∈ Cat → (2nd𝐼) = (𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧))))
1413opeq2d 4809 . . 3 (𝐶 ∈ Cat → ⟨( I ↾ (Base‘𝐶)), (2nd𝐼)⟩ = ⟨( I ↾ (Base‘𝐶)), (𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧)))⟩)
155, 14eqtr4d 2864 . 2 (𝐶 ∈ Cat → 𝐼 = ⟨( I ↾ (Base‘𝐶)), (2nd𝐼)⟩)
16 f1oi 6649 . . . . 5 ( I ↾ (Base‘𝐶)):(Base‘𝐶)–1-1-onto→(Base‘𝐶)
17 f1of 6612 . . . . 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 6649 . . . . . . . . . 10 ( I ↾ ((Hom ‘𝐶)‘𝑧)):((Hom ‘𝐶)‘𝑧)–1-1-onto→((Hom ‘𝐶)‘𝑧)
20 f1of 6612 . . . . . . . . . 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 6680 . . . . . . . . . 10 ((Hom ‘𝐶)‘𝑧) ∈ V
2322, 22elmap 8425 . . . . . . . . 9 (( I ↾ ((Hom ‘𝐶)‘𝑧)) ∈ (((Hom ‘𝐶)‘𝑧) ↑m ((Hom ‘𝐶)‘𝑧)) ↔ ( I ↾ ((Hom ‘𝐶)‘𝑧)):((Hom ‘𝐶)‘𝑧)⟶((Hom ‘𝐶)‘𝑧))
2421, 23mpbir 232 . . . . . . . 8 ( I ↾ ((Hom ‘𝐶)‘𝑧)) ∈ (((Hom ‘𝐶)‘𝑧) ↑m ((Hom ‘𝐶)‘𝑧))
25 xp1st 7712 . . . . . . . . . . . . . 14 (𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶)) → (1st𝑧) ∈ (Base‘𝐶))
2625adantl 482 . . . . . . . . . . . . 13 ((𝐶 ∈ Cat ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) → (1st𝑧) ∈ (Base‘𝐶))
27 fvresi 6931 . . . . . . . . . . . . 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 7713 . . . . . . . . . . . . . 14 (𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶)) → (2nd𝑧) ∈ (Base‘𝐶))
3029adantl 482 . . . . . . . . . . . . 13 ((𝐶 ∈ Cat ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) → (2nd𝑧) ∈ (Base‘𝐶))
31 fvresi 6931 . . . . . . . . . . . . 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 7166 . . . . . . . . . . 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 7151 . . . . . . . . . . 11 ((1st𝑧)(Hom ‘𝐶)(2nd𝑧)) = ((Hom ‘𝐶)‘⟨(1st𝑧), (2nd𝑧)⟩)
3533, 34syl6eq 2877 . . . . . . . . . 10 ((𝐶 ∈ Cat ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((( I ↾ (Base‘𝐶))‘(1st𝑧))(Hom ‘𝐶)(( I ↾ (Base‘𝐶))‘(2nd𝑧))) = ((Hom ‘𝐶)‘⟨(1st𝑧), (2nd𝑧)⟩))
36 1st2nd2 7719 . . . . . . . . . . . 12 (𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶)) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
3736adantl 482 . . . . . . . . . . 11 ((𝐶 ∈ Cat ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
3837fveq2d 6671 . . . . . . . . . 10 ((𝐶 ∈ Cat ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Hom ‘𝐶)‘𝑧) = ((Hom ‘𝐶)‘⟨(1st𝑧), (2nd𝑧)⟩))
3935, 38eqtr4d 2864 . . . . . . . . 9 ((𝐶 ∈ Cat ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((( I ↾ (Base‘𝐶))‘(1st𝑧))(Hom ‘𝐶)(( I ↾ (Base‘𝐶))‘(2nd𝑧))) = ((Hom ‘𝐶)‘𝑧))
4039oveq1d 7163 . . . . . . . 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, 40eleqtrrid 2925 . . . . . . 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 3187 . . . . . 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 8488 . . . . . . 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 235 . . . . 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 2918 . . . 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 2826 . . . . . . . . 9 (Id‘𝐶) = (Id‘𝐶)
48 simpl 483 . . . . . . . . 9 ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐶 ∈ Cat)
49 simpr 485 . . . . . . . . 9 ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
502, 4, 47, 48, 49catidcl 16943 . . . . . . . 8 ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥))
51 fvresi 6931 . . . . . . . 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 17137 . . . . . . . 8 ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑥(2nd𝐼)𝑥) = ( I ↾ (𝑥(Hom ‘𝐶)𝑥)))
5453fveq1d 6669 . . . . . . 7 ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝑥(2nd𝐼)𝑥)‘((Id‘𝐶)‘𝑥)) = (( I ↾ (𝑥(Hom ‘𝐶)𝑥))‘((Id‘𝐶)‘𝑥)))
55 fvresi 6931 . . . . . . . . 9 (𝑥 ∈ (Base‘𝐶) → (( I ↾ (Base‘𝐶))‘𝑥) = 𝑥)
5655adantl 482 . . . . . . . 8 ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → (( I ↾ (Base‘𝐶))‘𝑥) = 𝑥)
5756fveq2d 6671 . . . . . . 7 ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘(( I ↾ (Base‘𝐶))‘𝑥)) = ((Id‘𝐶)‘𝑥))
5852, 54, 573eqtr4d 2871 . . . . . 6 ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝑥(2nd𝐼)𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐶)‘(( I ↾ (Base‘𝐶))‘𝑥)))
59 eqid 2826 . . . . . . . . . . 11 (comp‘𝐶) = (comp‘𝐶)
6048ad2antrr 722 . . . . . . . . . . 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 722 . . . . . . . . . . 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 773 . . . . . . . . . . 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 774 . . . . . . . . . . 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 767 . . . . . . . . . . 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 769 . . . . . . . . . . 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 16946 . . . . . . . . . 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 6931 . . . . . . . . . 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 17137 . . . . . . . . . 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 6669 . . . . . . . . 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 6931 . . . . . . . . . . . . 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 4810 . . . . . . . . . . 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 6931 . . . . . . . . . . . 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 7166 . . . . . . . . . 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 17138 . . . . . . . . . 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 17138 . . . . . . . . . 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 7169 . . . . . . . . 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 2871 . . . . . . . 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 3196 . . . . . . 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 3196 . . . . . 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 512 . . . . 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 3187 . . . 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 17124 . . . 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 1336 . . 3 (𝐶 ∈ Cat → ( I ↾ (Base‘𝐶))(𝐶 Func 𝐶)(2nd𝐼))
88 df-br 5064 . . 3 (( I ↾ (Base‘𝐶))(𝐶 Func 𝐶)(2nd𝐼) ↔ ⟨( I ↾ (Base‘𝐶)), (2nd𝐼)⟩ ∈ (𝐶 Func 𝐶))
8987, 88sylib 219 . 2 (𝐶 ∈ Cat → ⟨( I ↾ (Base‘𝐶)), (2nd𝐼)⟩ ∈ (𝐶 Func 𝐶))
9015, 89eqeltrd 2918 1 (𝐶 ∈ Cat → 𝐼 ∈ (𝐶 Func 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  wral 3143  Vcvv 3500  cop 4570   class class class wbr 5063  cmpt 5143   I cid 5458   × cxp 5552  cres 5556  wf 6348  1-1-ontowf1o 6351  cfv 6352  (class class class)co 7148  1st c1st 7678  2nd c2nd 7679  m cmap 8396  Xcixp 8450  Basecbs 16473  Hom chom 16566  compcco 16567  Catccat 16925  Idccid 16926   Func cfunc 17114  idfunccidfu 17115
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7680  df-2nd 7681  df-map 8398  df-ixp 8451  df-cat 16929  df-cid 16930  df-func 17118  df-idfu 17119
This theorem is referenced by:  cofulid  17150  cofurid  17151  idffth  17193  ressffth  17198  catccatid  17352  curf2ndf  17487
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