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Theorem funcres 17941
Description: A functor restricted to a subcategory is a functor. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
funcres.f (𝜑𝐹 ∈ (𝐶 Func 𝐷))
funcres.h (𝜑𝐻 ∈ (Subcat‘𝐶))
Assertion
Ref Expression
funcres (𝜑 → (𝐹f 𝐻) ∈ ((𝐶cat 𝐻) Func 𝐷))

Proof of Theorem funcres
Dummy variables 𝑓 𝑔 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funcres.f . . . 4 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
2 funcres.h . . . 4 (𝜑𝐻 ∈ (Subcat‘𝐶))
31, 2resfval 17937 . . 3 (𝜑 → (𝐹f 𝐻) = ⟨((1st𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)))⟩)
43fveq2d 6910 . . . . 5 (𝜑 → (2nd ‘(𝐹f 𝐻)) = (2nd ‘⟨((1st𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)))⟩))
5 fvex 6919 . . . . . . 7 (1st𝐹) ∈ V
65resex 6047 . . . . . 6 ((1st𝐹) ↾ dom dom 𝐻) ∈ V
7 dmexg 7923 . . . . . . 7 (𝐻 ∈ (Subcat‘𝐶) → dom 𝐻 ∈ V)
8 mptexg 7241 . . . . . . 7 (dom 𝐻 ∈ V → (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧))) ∈ V)
92, 7, 83syl 18 . . . . . 6 (𝜑 → (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧))) ∈ V)
10 op2ndg 8027 . . . . . 6 ((((1st𝐹) ↾ dom dom 𝐻) ∈ V ∧ (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧))) ∈ V) → (2nd ‘⟨((1st𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)))⟩) = (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧))))
116, 9, 10sylancr 587 . . . . 5 (𝜑 → (2nd ‘⟨((1st𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)))⟩) = (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧))))
124, 11eqtrd 2777 . . . 4 (𝜑 → (2nd ‘(𝐹f 𝐻)) = (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧))))
1312opeq2d 4880 . . 3 (𝜑 → ⟨((1st𝐹) ↾ dom dom 𝐻), (2nd ‘(𝐹f 𝐻))⟩ = ⟨((1st𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)))⟩)
143, 13eqtr4d 2780 . 2 (𝜑 → (𝐹f 𝐻) = ⟨((1st𝐹) ↾ dom dom 𝐻), (2nd ‘(𝐹f 𝐻))⟩)
15 eqid 2737 . . . 4 (Base‘(𝐶cat 𝐻)) = (Base‘(𝐶cat 𝐻))
16 eqid 2737 . . . 4 (Base‘𝐷) = (Base‘𝐷)
17 eqid 2737 . . . 4 (Hom ‘(𝐶cat 𝐻)) = (Hom ‘(𝐶cat 𝐻))
18 eqid 2737 . . . 4 (Hom ‘𝐷) = (Hom ‘𝐷)
19 eqid 2737 . . . 4 (Id‘(𝐶cat 𝐻)) = (Id‘(𝐶cat 𝐻))
20 eqid 2737 . . . 4 (Id‘𝐷) = (Id‘𝐷)
21 eqid 2737 . . . 4 (comp‘(𝐶cat 𝐻)) = (comp‘(𝐶cat 𝐻))
22 eqid 2737 . . . 4 (comp‘𝐷) = (comp‘𝐷)
23 eqid 2737 . . . . 5 (𝐶cat 𝐻) = (𝐶cat 𝐻)
2423, 2subccat 17893 . . . 4 (𝜑 → (𝐶cat 𝐻) ∈ Cat)
25 funcrcl 17908 . . . . . 6 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
261, 25syl 17 . . . . 5 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
2726simprd 495 . . . 4 (𝜑𝐷 ∈ Cat)
28 eqid 2737 . . . . . . 7 (Base‘𝐶) = (Base‘𝐶)
29 relfunc 17907 . . . . . . . 8 Rel (𝐶 Func 𝐷)
30 1st2ndbr 8067 . . . . . . . 8 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
3129, 1, 30sylancr 587 . . . . . . 7 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
3228, 16, 31funcf1 17911 . . . . . 6 (𝜑 → (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷))
33 eqidd 2738 . . . . . . . 8 (𝜑 → dom dom 𝐻 = dom dom 𝐻)
342, 33subcfn 17886 . . . . . . 7 (𝜑𝐻 Fn (dom dom 𝐻 × dom dom 𝐻))
352, 34, 28subcss1 17887 . . . . . 6 (𝜑 → dom dom 𝐻 ⊆ (Base‘𝐶))
3632, 35fssresd 6775 . . . . 5 (𝜑 → ((1st𝐹) ↾ dom dom 𝐻):dom dom 𝐻⟶(Base‘𝐷))
3726simpld 494 . . . . . . 7 (𝜑𝐶 ∈ Cat)
3823, 28, 37, 34, 35rescbas 17873 . . . . . 6 (𝜑 → dom dom 𝐻 = (Base‘(𝐶cat 𝐻)))
3938feq2d 6722 . . . . 5 (𝜑 → (((1st𝐹) ↾ dom dom 𝐻):dom dom 𝐻⟶(Base‘𝐷) ↔ ((1st𝐹) ↾ dom dom 𝐻):(Base‘(𝐶cat 𝐻))⟶(Base‘𝐷)))
4036, 39mpbid 232 . . . 4 (𝜑 → ((1st𝐹) ↾ dom dom 𝐻):(Base‘(𝐶cat 𝐻))⟶(Base‘𝐷))
41 fvex 6919 . . . . . . 7 ((2nd𝐹)‘𝑧) ∈ V
4241resex 6047 . . . . . 6 (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)) ∈ V
43 eqid 2737 . . . . . 6 (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧))) = (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)))
4442, 43fnmpti 6711 . . . . 5 (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧))) Fn dom 𝐻
4512eqcomd 2743 . . . . . 6 (𝜑 → (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧))) = (2nd ‘(𝐹f 𝐻)))
46 fndm 6671 . . . . . . . 8 (𝐻 Fn (dom dom 𝐻 × dom dom 𝐻) → dom 𝐻 = (dom dom 𝐻 × dom dom 𝐻))
4734, 46syl 17 . . . . . . 7 (𝜑 → dom 𝐻 = (dom dom 𝐻 × dom dom 𝐻))
4838sqxpeqd 5717 . . . . . . 7 (𝜑 → (dom dom 𝐻 × dom dom 𝐻) = ((Base‘(𝐶cat 𝐻)) × (Base‘(𝐶cat 𝐻))))
4947, 48eqtrd 2777 . . . . . 6 (𝜑 → dom 𝐻 = ((Base‘(𝐶cat 𝐻)) × (Base‘(𝐶cat 𝐻))))
5045, 49fneq12d 6663 . . . . 5 (𝜑 → ((𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧))) Fn dom 𝐻 ↔ (2nd ‘(𝐹f 𝐻)) Fn ((Base‘(𝐶cat 𝐻)) × (Base‘(𝐶cat 𝐻)))))
5144, 50mpbii 233 . . . 4 (𝜑 → (2nd ‘(𝐹f 𝐻)) Fn ((Base‘(𝐶cat 𝐻)) × (Base‘(𝐶cat 𝐻))))
52 eqid 2737 . . . . . . . 8 (Hom ‘𝐶) = (Hom ‘𝐶)
5331adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
5435adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → dom dom 𝐻 ⊆ (Base‘𝐶))
55 simprl 771 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → 𝑥 ∈ (Base‘(𝐶cat 𝐻)))
5638adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → dom dom 𝐻 = (Base‘(𝐶cat 𝐻)))
5755, 56eleqtrrd 2844 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → 𝑥 ∈ dom dom 𝐻)
5854, 57sseldd 3984 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → 𝑥 ∈ (Base‘𝐶))
59 simprr 773 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → 𝑦 ∈ (Base‘(𝐶cat 𝐻)))
6059, 56eleqtrrd 2844 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → 𝑦 ∈ dom dom 𝐻)
6154, 60sseldd 3984 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → 𝑦 ∈ (Base‘𝐶))
6228, 52, 18, 53, 58, 61funcf2 17913 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → (𝑥(2nd𝐹)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
632adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → 𝐻 ∈ (Subcat‘𝐶))
6434adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → 𝐻 Fn (dom dom 𝐻 × dom dom 𝐻))
6563, 64, 52, 57, 60subcss2 17888 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → (𝑥𝐻𝑦) ⊆ (𝑥(Hom ‘𝐶)𝑦))
6662, 65fssresd 6775 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → ((𝑥(2nd𝐹)𝑦) ↾ (𝑥𝐻𝑦)):(𝑥𝐻𝑦)⟶(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
671adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → 𝐹 ∈ (𝐶 Func 𝐷))
6867, 63, 64, 57, 60resf2nd 17940 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → (𝑥(2nd ‘(𝐹f 𝐻))𝑦) = ((𝑥(2nd𝐹)𝑦) ↾ (𝑥𝐻𝑦)))
6968feq1d 6720 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → ((𝑥(2nd ‘(𝐹f 𝐻))𝑦):(𝑥𝐻𝑦)⟶(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)) ↔ ((𝑥(2nd𝐹)𝑦) ↾ (𝑥𝐻𝑦)):(𝑥𝐻𝑦)⟶(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦))))
7066, 69mpbird 257 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → (𝑥(2nd ‘(𝐹f 𝐻))𝑦):(𝑥𝐻𝑦)⟶(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
7123, 28, 37, 34, 35reschom 17874 . . . . . . . 8 (𝜑𝐻 = (Hom ‘(𝐶cat 𝐻)))
7271adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → 𝐻 = (Hom ‘(𝐶cat 𝐻)))
7372oveqd 7448 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → (𝑥𝐻𝑦) = (𝑥(Hom ‘(𝐶cat 𝐻))𝑦))
7457fvresd 6926 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → (((1st𝐹) ↾ dom dom 𝐻)‘𝑥) = ((1st𝐹)‘𝑥))
7560fvresd 6926 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → (((1st𝐹) ↾ dom dom 𝐻)‘𝑦) = ((1st𝐹)‘𝑦))
7674, 75oveq12d 7449 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → ((((1st𝐹) ↾ dom dom 𝐻)‘𝑥)(Hom ‘𝐷)(((1st𝐹) ↾ dom dom 𝐻)‘𝑦)) = (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
7776eqcomd 2743 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)) = ((((1st𝐹) ↾ dom dom 𝐻)‘𝑥)(Hom ‘𝐷)(((1st𝐹) ↾ dom dom 𝐻)‘𝑦)))
7873, 77feq23d 6731 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → ((𝑥(2nd ‘(𝐹f 𝐻))𝑦):(𝑥𝐻𝑦)⟶(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)) ↔ (𝑥(2nd ‘(𝐹f 𝐻))𝑦):(𝑥(Hom ‘(𝐶cat 𝐻))𝑦)⟶((((1st𝐹) ↾ dom dom 𝐻)‘𝑥)(Hom ‘𝐷)(((1st𝐹) ↾ dom dom 𝐻)‘𝑦))))
7970, 78mpbid 232 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → (𝑥(2nd ‘(𝐹f 𝐻))𝑦):(𝑥(Hom ‘(𝐶cat 𝐻))𝑦)⟶((((1st𝐹) ↾ dom dom 𝐻)‘𝑥)(Hom ‘𝐷)(((1st𝐹) ↾ dom dom 𝐻)‘𝑦)))
801adantr 480 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘(𝐶cat 𝐻))) → 𝐹 ∈ (𝐶 Func 𝐷))
812adantr 480 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘(𝐶cat 𝐻))) → 𝐻 ∈ (Subcat‘𝐶))
8234adantr 480 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘(𝐶cat 𝐻))) → 𝐻 Fn (dom dom 𝐻 × dom dom 𝐻))
8338eleq2d 2827 . . . . . . . 8 (𝜑 → (𝑥 ∈ dom dom 𝐻𝑥 ∈ (Base‘(𝐶cat 𝐻))))
8483biimpar 477 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘(𝐶cat 𝐻))) → 𝑥 ∈ dom dom 𝐻)
8580, 81, 82, 84, 84resf2nd 17940 . . . . . 6 ((𝜑𝑥 ∈ (Base‘(𝐶cat 𝐻))) → (𝑥(2nd ‘(𝐹f 𝐻))𝑥) = ((𝑥(2nd𝐹)𝑥) ↾ (𝑥𝐻𝑥)))
86 eqid 2737 . . . . . . . 8 (Id‘𝐶) = (Id‘𝐶)
8723, 81, 82, 86, 84subcid 17892 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘(𝐶cat 𝐻))) → ((Id‘𝐶)‘𝑥) = ((Id‘(𝐶cat 𝐻))‘𝑥))
8887eqcomd 2743 . . . . . 6 ((𝜑𝑥 ∈ (Base‘(𝐶cat 𝐻))) → ((Id‘(𝐶cat 𝐻))‘𝑥) = ((Id‘𝐶)‘𝑥))
8985, 88fveq12d 6913 . . . . 5 ((𝜑𝑥 ∈ (Base‘(𝐶cat 𝐻))) → ((𝑥(2nd ‘(𝐹f 𝐻))𝑥)‘((Id‘(𝐶cat 𝐻))‘𝑥)) = (((𝑥(2nd𝐹)𝑥) ↾ (𝑥𝐻𝑥))‘((Id‘𝐶)‘𝑥)))
9031adantr 480 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘(𝐶cat 𝐻))) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
9138, 35eqsstrrd 4019 . . . . . . . 8 (𝜑 → (Base‘(𝐶cat 𝐻)) ⊆ (Base‘𝐶))
9291sselda 3983 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘(𝐶cat 𝐻))) → 𝑥 ∈ (Base‘𝐶))
9328, 86, 20, 90, 92funcid 17915 . . . . . 6 ((𝜑𝑥 ∈ (Base‘(𝐶cat 𝐻))) → ((𝑥(2nd𝐹)𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐷)‘((1st𝐹)‘𝑥)))
9481, 82, 84, 86subcidcl 17889 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘(𝐶cat 𝐻))) → ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥))
9594fvresd 6926 . . . . . 6 ((𝜑𝑥 ∈ (Base‘(𝐶cat 𝐻))) → (((𝑥(2nd𝐹)𝑥) ↾ (𝑥𝐻𝑥))‘((Id‘𝐶)‘𝑥)) = ((𝑥(2nd𝐹)𝑥)‘((Id‘𝐶)‘𝑥)))
9684fvresd 6926 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘(𝐶cat 𝐻))) → (((1st𝐹) ↾ dom dom 𝐻)‘𝑥) = ((1st𝐹)‘𝑥))
9796fveq2d 6910 . . . . . 6 ((𝜑𝑥 ∈ (Base‘(𝐶cat 𝐻))) → ((Id‘𝐷)‘(((1st𝐹) ↾ dom dom 𝐻)‘𝑥)) = ((Id‘𝐷)‘((1st𝐹)‘𝑥)))
9893, 95, 973eqtr4d 2787 . . . . 5 ((𝜑𝑥 ∈ (Base‘(𝐶cat 𝐻))) → (((𝑥(2nd𝐹)𝑥) ↾ (𝑥𝐻𝑥))‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐷)‘(((1st𝐹) ↾ dom dom 𝐻)‘𝑥)))
9989, 98eqtrd 2777 . . . 4 ((𝜑𝑥 ∈ (Base‘(𝐶cat 𝐻))) → ((𝑥(2nd ‘(𝐹f 𝐻))𝑥)‘((Id‘(𝐶cat 𝐻))‘𝑥)) = ((Id‘𝐷)‘(((1st𝐹) ↾ dom dom 𝐻)‘𝑥)))
10023ad2ant1 1134 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → 𝐻 ∈ (Subcat‘𝐶))
101343ad2ant1 1134 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → 𝐻 Fn (dom dom 𝐻 × dom dom 𝐻))
102 simp21 1207 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → 𝑥 ∈ (Base‘(𝐶cat 𝐻)))
103383ad2ant1 1134 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → dom dom 𝐻 = (Base‘(𝐶cat 𝐻)))
104102, 103eleqtrrd 2844 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → 𝑥 ∈ dom dom 𝐻)
105 eqid 2737 . . . . . . . 8 (comp‘𝐶) = (comp‘𝐶)
106 simp22 1208 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → 𝑦 ∈ (Base‘(𝐶cat 𝐻)))
107106, 103eleqtrrd 2844 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → 𝑦 ∈ dom dom 𝐻)
108 simp23 1209 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → 𝑧 ∈ (Base‘(𝐶cat 𝐻)))
109108, 103eleqtrrd 2844 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → 𝑧 ∈ dom dom 𝐻)
110 simp3l 1202 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → 𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦))
111713ad2ant1 1134 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → 𝐻 = (Hom ‘(𝐶cat 𝐻)))
112111oveqd 7448 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → (𝑥𝐻𝑦) = (𝑥(Hom ‘(𝐶cat 𝐻))𝑦))
113110, 112eleqtrrd 2844 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → 𝑓 ∈ (𝑥𝐻𝑦))
114 simp3r 1203 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))
115111oveqd 7448 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → (𝑦𝐻𝑧) = (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))
116114, 115eleqtrrd 2844 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → 𝑔 ∈ (𝑦𝐻𝑧))
117100, 101, 104, 105, 107, 109, 113, 116subccocl 17890 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐻𝑧))
118117fvresd 6926 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → (((𝑥(2nd𝐹)𝑧) ↾ (𝑥𝐻𝑧))‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = ((𝑥(2nd𝐹)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))
119313ad2ant1 1134 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
120353ad2ant1 1134 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → dom dom 𝐻 ⊆ (Base‘𝐶))
121120, 104sseldd 3984 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → 𝑥 ∈ (Base‘𝐶))
122120, 107sseldd 3984 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → 𝑦 ∈ (Base‘𝐶))
123120, 109sseldd 3984 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → 𝑧 ∈ (Base‘𝐶))
124100, 101, 52, 104, 107subcss2 17888 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → (𝑥𝐻𝑦) ⊆ (𝑥(Hom ‘𝐶)𝑦))
125124, 113sseldd 3984 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
126100, 101, 52, 107, 109subcss2 17888 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → (𝑦𝐻𝑧) ⊆ (𝑦(Hom ‘𝐶)𝑧))
127126, 116sseldd 3984 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))
12828, 52, 105, 22, 119, 121, 122, 123, 125, 127funcco 17916 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → ((𝑥(2nd𝐹)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦(2nd𝐹)𝑧)‘𝑔)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑥(2nd𝐹)𝑦)‘𝑓)))
129118, 128eqtrd 2777 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → (((𝑥(2nd𝐹)𝑧) ↾ (𝑥𝐻𝑧))‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦(2nd𝐹)𝑧)‘𝑔)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑥(2nd𝐹)𝑦)‘𝑓)))
13013ad2ant1 1134 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → 𝐹 ∈ (𝐶 Func 𝐷))
131130, 100, 101, 104, 109resf2nd 17940 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → (𝑥(2nd ‘(𝐹f 𝐻))𝑧) = ((𝑥(2nd𝐹)𝑧) ↾ (𝑥𝐻𝑧)))
13223, 28, 37, 34, 35, 105rescco 17876 . . . . . . . . . 10 (𝜑 → (comp‘𝐶) = (comp‘(𝐶cat 𝐻)))
1331323ad2ant1 1134 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → (comp‘𝐶) = (comp‘(𝐶cat 𝐻)))
134133eqcomd 2743 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → (comp‘(𝐶cat 𝐻)) = (comp‘𝐶))
135134oveqd 7448 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → (⟨𝑥, 𝑦⟩(comp‘(𝐶cat 𝐻))𝑧) = (⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧))
136135oveqd 7448 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘(𝐶cat 𝐻))𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))
137131, 136fveq12d 6913 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → ((𝑥(2nd ‘(𝐹f 𝐻))𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘(𝐶cat 𝐻))𝑧)𝑓)) = (((𝑥(2nd𝐹)𝑧) ↾ (𝑥𝐻𝑧))‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))
138104fvresd 6926 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → (((1st𝐹) ↾ dom dom 𝐻)‘𝑥) = ((1st𝐹)‘𝑥))
139107fvresd 6926 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → (((1st𝐹) ↾ dom dom 𝐻)‘𝑦) = ((1st𝐹)‘𝑦))
140138, 139opeq12d 4881 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → ⟨(((1st𝐹) ↾ dom dom 𝐻)‘𝑥), (((1st𝐹) ↾ dom dom 𝐻)‘𝑦)⟩ = ⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑦)⟩)
141109fvresd 6926 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → (((1st𝐹) ↾ dom dom 𝐻)‘𝑧) = ((1st𝐹)‘𝑧))
142140, 141oveq12d 7449 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → (⟨(((1st𝐹) ↾ dom dom 𝐻)‘𝑥), (((1st𝐹) ↾ dom dom 𝐻)‘𝑦)⟩(comp‘𝐷)(((1st𝐹) ↾ dom dom 𝐻)‘𝑧)) = (⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐹)‘𝑧)))
143130, 100, 101, 107, 109resf2nd 17940 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → (𝑦(2nd ‘(𝐹f 𝐻))𝑧) = ((𝑦(2nd𝐹)𝑧) ↾ (𝑦𝐻𝑧)))
144143fveq1d 6908 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → ((𝑦(2nd ‘(𝐹f 𝐻))𝑧)‘𝑔) = (((𝑦(2nd𝐹)𝑧) ↾ (𝑦𝐻𝑧))‘𝑔))
145116fvresd 6926 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → (((𝑦(2nd𝐹)𝑧) ↾ (𝑦𝐻𝑧))‘𝑔) = ((𝑦(2nd𝐹)𝑧)‘𝑔))
146144, 145eqtrd 2777 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → ((𝑦(2nd ‘(𝐹f 𝐻))𝑧)‘𝑔) = ((𝑦(2nd𝐹)𝑧)‘𝑔))
147130, 100, 101, 104, 107resf2nd 17940 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → (𝑥(2nd ‘(𝐹f 𝐻))𝑦) = ((𝑥(2nd𝐹)𝑦) ↾ (𝑥𝐻𝑦)))
148147fveq1d 6908 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → ((𝑥(2nd ‘(𝐹f 𝐻))𝑦)‘𝑓) = (((𝑥(2nd𝐹)𝑦) ↾ (𝑥𝐻𝑦))‘𝑓))
149113fvresd 6926 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → (((𝑥(2nd𝐹)𝑦) ↾ (𝑥𝐻𝑦))‘𝑓) = ((𝑥(2nd𝐹)𝑦)‘𝑓))
150148, 149eqtrd 2777 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → ((𝑥(2nd ‘(𝐹f 𝐻))𝑦)‘𝑓) = ((𝑥(2nd𝐹)𝑦)‘𝑓))
151142, 146, 150oveq123d 7452 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → (((𝑦(2nd ‘(𝐹f 𝐻))𝑧)‘𝑔)(⟨(((1st𝐹) ↾ dom dom 𝐻)‘𝑥), (((1st𝐹) ↾ dom dom 𝐻)‘𝑦)⟩(comp‘𝐷)(((1st𝐹) ↾ dom dom 𝐻)‘𝑧))((𝑥(2nd ‘(𝐹f 𝐻))𝑦)‘𝑓)) = (((𝑦(2nd𝐹)𝑧)‘𝑔)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑥(2nd𝐹)𝑦)‘𝑓)))
152129, 137, 1513eqtr4d 2787 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶cat 𝐻))𝑧))) → ((𝑥(2nd ‘(𝐹f 𝐻))𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘(𝐶cat 𝐻))𝑧)𝑓)) = (((𝑦(2nd ‘(𝐹f 𝐻))𝑧)‘𝑔)(⟨(((1st𝐹) ↾ dom dom 𝐻)‘𝑥), (((1st𝐹) ↾ dom dom 𝐻)‘𝑦)⟩(comp‘𝐷)(((1st𝐹) ↾ dom dom 𝐻)‘𝑧))((𝑥(2nd ‘(𝐹f 𝐻))𝑦)‘𝑓)))
15315, 16, 17, 18, 19, 20, 21, 22, 24, 27, 40, 51, 79, 99, 152isfuncd 17910 . . 3 (𝜑 → ((1st𝐹) ↾ dom dom 𝐻)((𝐶cat 𝐻) Func 𝐷)(2nd ‘(𝐹f 𝐻)))
154 df-br 5144 . . 3 (((1st𝐹) ↾ dom dom 𝐻)((𝐶cat 𝐻) Func 𝐷)(2nd ‘(𝐹f 𝐻)) ↔ ⟨((1st𝐹) ↾ dom dom 𝐻), (2nd ‘(𝐹f 𝐻))⟩ ∈ ((𝐶cat 𝐻) Func 𝐷))
155153, 154sylib 218 . 2 (𝜑 → ⟨((1st𝐹) ↾ dom dom 𝐻), (2nd ‘(𝐹f 𝐻))⟩ ∈ ((𝐶cat 𝐻) Func 𝐷))
15614, 155eqeltrd 2841 1 (𝜑 → (𝐹f 𝐻) ∈ ((𝐶cat 𝐻) Func 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  Vcvv 3480  wss 3951  cop 4632   class class class wbr 5143  cmpt 5225   × cxp 5683  dom cdm 5685  cres 5687  Rel wrel 5690   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  Basecbs 17247  Hom chom 17308  compcco 17309  Catccat 17707  Idccid 17708  cat cresc 17852  Subcatcsubc 17853   Func cfunc 17899  f cresf 17902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-hom 17321  df-cco 17322  df-cat 17711  df-cid 17712  df-homf 17713  df-ssc 17854  df-resc 17855  df-subc 17856  df-func 17903  df-resf 17906
This theorem is referenced by:  funcrngcsetc  20640  funcringcsetc  20674
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