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Theorem funcres 17158
 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 17154 . . 3 (𝜑 → (𝐹f 𝐻) = ⟨((1st𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)))⟩)
43fveq2d 6649 . . . . 5 (𝜑 → (2nd ‘(𝐹f 𝐻)) = (2nd ‘⟨((1st𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)))⟩))
5 fvex 6658 . . . . . . 7 (1st𝐹) ∈ V
65resex 5866 . . . . . 6 ((1st𝐹) ↾ dom dom 𝐻) ∈ V
7 dmexg 7594 . . . . . . 7 (𝐻 ∈ (Subcat‘𝐶) → dom 𝐻 ∈ V)
8 mptexg 6961 . . . . . . 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 7684 . . . . . 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 590 . . . . 5 (𝜑 → (2nd ‘⟨((1st𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)))⟩) = (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧))))
124, 11eqtrd 2833 . . . 4 (𝜑 → (2nd ‘(𝐹f 𝐻)) = (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧))))
1312opeq2d 4772 . . 3 (𝜑 → ⟨((1st𝐹) ↾ dom dom 𝐻), (2nd ‘(𝐹f 𝐻))⟩ = ⟨((1st𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)))⟩)
143, 13eqtr4d 2836 . 2 (𝜑 → (𝐹f 𝐻) = ⟨((1st𝐹) ↾ dom dom 𝐻), (2nd ‘(𝐹f 𝐻))⟩)
15 eqid 2798 . . . 4 (Base‘(𝐶cat 𝐻)) = (Base‘(𝐶cat 𝐻))
16 eqid 2798 . . . 4 (Base‘𝐷) = (Base‘𝐷)
17 eqid 2798 . . . 4 (Hom ‘(𝐶cat 𝐻)) = (Hom ‘(𝐶cat 𝐻))
18 eqid 2798 . . . 4 (Hom ‘𝐷) = (Hom ‘𝐷)
19 eqid 2798 . . . 4 (Id‘(𝐶cat 𝐻)) = (Id‘(𝐶cat 𝐻))
20 eqid 2798 . . . 4 (Id‘𝐷) = (Id‘𝐷)
21 eqid 2798 . . . 4 (comp‘(𝐶cat 𝐻)) = (comp‘(𝐶cat 𝐻))
22 eqid 2798 . . . 4 (comp‘𝐷) = (comp‘𝐷)
23 eqid 2798 . . . . 5 (𝐶cat 𝐻) = (𝐶cat 𝐻)
2423, 2subccat 17110 . . . 4 (𝜑 → (𝐶cat 𝐻) ∈ Cat)
25 funcrcl 17125 . . . . . 6 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
261, 25syl 17 . . . . 5 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
2726simprd 499 . . . 4 (𝜑𝐷 ∈ Cat)
28 eqid 2798 . . . . . . 7 (Base‘𝐶) = (Base‘𝐶)
29 relfunc 17124 . . . . . . . 8 Rel (𝐶 Func 𝐷)
30 1st2ndbr 7723 . . . . . . . 8 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
3129, 1, 30sylancr 590 . . . . . . 7 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
3228, 16, 31funcf1 17128 . . . . . 6 (𝜑 → (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷))
33 eqidd 2799 . . . . . . . 8 (𝜑 → dom dom 𝐻 = dom dom 𝐻)
342, 33subcfn 17103 . . . . . . 7 (𝜑𝐻 Fn (dom dom 𝐻 × dom dom 𝐻))
352, 34, 28subcss1 17104 . . . . . 6 (𝜑 → dom dom 𝐻 ⊆ (Base‘𝐶))
3632, 35fssresd 6519 . . . . 5 (𝜑 → ((1st𝐹) ↾ dom dom 𝐻):dom dom 𝐻⟶(Base‘𝐷))
3726simpld 498 . . . . . . 7 (𝜑𝐶 ∈ Cat)
3823, 28, 37, 34, 35rescbas 17091 . . . . . 6 (𝜑 → dom dom 𝐻 = (Base‘(𝐶cat 𝐻)))
3938feq2d 6473 . . . . 5 (𝜑 → (((1st𝐹) ↾ dom dom 𝐻):dom dom 𝐻⟶(Base‘𝐷) ↔ ((1st𝐹) ↾ dom dom 𝐻):(Base‘(𝐶cat 𝐻))⟶(Base‘𝐷)))
4036, 39mpbid 235 . . . 4 (𝜑 → ((1st𝐹) ↾ dom dom 𝐻):(Base‘(𝐶cat 𝐻))⟶(Base‘𝐷))
41 fvex 6658 . . . . . . 7 ((2nd𝐹)‘𝑧) ∈ V
4241resex 5866 . . . . . 6 (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)) ∈ V
43 eqid 2798 . . . . . 6 (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧))) = (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)))
4442, 43fnmpti 6463 . . . . 5 (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧))) Fn dom 𝐻
4512eqcomd 2804 . . . . . 6 (𝜑 → (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧))) = (2nd ‘(𝐹f 𝐻)))
46 fndm 6425 . . . . . . . 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 5551 . . . . . . 7 (𝜑 → (dom dom 𝐻 × dom dom 𝐻) = ((Base‘(𝐶cat 𝐻)) × (Base‘(𝐶cat 𝐻))))
4947, 48eqtrd 2833 . . . . . 6 (𝜑 → dom 𝐻 = ((Base‘(𝐶cat 𝐻)) × (Base‘(𝐶cat 𝐻))))
5045, 49fneq12d 6418 . . . . 5 (𝜑 → ((𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧))) Fn dom 𝐻 ↔ (2nd ‘(𝐹f 𝐻)) Fn ((Base‘(𝐶cat 𝐻)) × (Base‘(𝐶cat 𝐻)))))
5144, 50mpbii 236 . . . 4 (𝜑 → (2nd ‘(𝐹f 𝐻)) Fn ((Base‘(𝐶cat 𝐻)) × (Base‘(𝐶cat 𝐻))))
52 eqid 2798 . . . . . . . 8 (Hom ‘𝐶) = (Hom ‘𝐶)
5331adantr 484 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
5435adantr 484 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → dom dom 𝐻 ⊆ (Base‘𝐶))
55 simprl 770 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → 𝑥 ∈ (Base‘(𝐶cat 𝐻)))
5638adantr 484 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → dom dom 𝐻 = (Base‘(𝐶cat 𝐻)))
5755, 56eleqtrrd 2893 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → 𝑥 ∈ dom dom 𝐻)
5854, 57sseldd 3916 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → 𝑥 ∈ (Base‘𝐶))
59 simprr 772 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → 𝑦 ∈ (Base‘(𝐶cat 𝐻)))
6059, 56eleqtrrd 2893 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → 𝑦 ∈ dom dom 𝐻)
6154, 60sseldd 3916 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → 𝑦 ∈ (Base‘𝐶))
6228, 52, 18, 53, 58, 61funcf2 17130 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → (𝑥(2nd𝐹)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
632adantr 484 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → 𝐻 ∈ (Subcat‘𝐶))
6434adantr 484 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → 𝐻 Fn (dom dom 𝐻 × dom dom 𝐻))
6563, 64, 52, 57, 60subcss2 17105 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → (𝑥𝐻𝑦) ⊆ (𝑥(Hom ‘𝐶)𝑦))
6662, 65fssresd 6519 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → ((𝑥(2nd𝐹)𝑦) ↾ (𝑥𝐻𝑦)):(𝑥𝐻𝑦)⟶(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
671adantr 484 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → 𝐹 ∈ (𝐶 Func 𝐷))
6867, 63, 64, 57, 60resf2nd 17157 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → (𝑥(2nd ‘(𝐹f 𝐻))𝑦) = ((𝑥(2nd𝐹)𝑦) ↾ (𝑥𝐻𝑦)))
6968feq1d 6472 . . . . . 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 260 . . . . 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 17092 . . . . . . . 8 (𝜑𝐻 = (Hom ‘(𝐶cat 𝐻)))
7271adantr 484 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → 𝐻 = (Hom ‘(𝐶cat 𝐻)))
7372oveqd 7152 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → (𝑥𝐻𝑦) = (𝑥(Hom ‘(𝐶cat 𝐻))𝑦))
7457fvresd 6665 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → (((1st𝐹) ↾ dom dom 𝐻)‘𝑥) = ((1st𝐹)‘𝑥))
7560fvresd 6665 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → (((1st𝐹) ↾ dom dom 𝐻)‘𝑦) = ((1st𝐹)‘𝑦))
7674, 75oveq12d 7153 . . . . . . 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 2804 . . . . . 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 6482 . . . . 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 235 . . . 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 484 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘(𝐶cat 𝐻))) → 𝐹 ∈ (𝐶 Func 𝐷))
812adantr 484 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘(𝐶cat 𝐻))) → 𝐻 ∈ (Subcat‘𝐶))
8234adantr 484 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘(𝐶cat 𝐻))) → 𝐻 Fn (dom dom 𝐻 × dom dom 𝐻))
8338eleq2d 2875 . . . . . . . 8 (𝜑 → (𝑥 ∈ dom dom 𝐻𝑥 ∈ (Base‘(𝐶cat 𝐻))))
8483biimpar 481 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘(𝐶cat 𝐻))) → 𝑥 ∈ dom dom 𝐻)
8580, 81, 82, 84, 84resf2nd 17157 . . . . . 6 ((𝜑𝑥 ∈ (Base‘(𝐶cat 𝐻))) → (𝑥(2nd ‘(𝐹f 𝐻))𝑥) = ((𝑥(2nd𝐹)𝑥) ↾ (𝑥𝐻𝑥)))
86 eqid 2798 . . . . . . . 8 (Id‘𝐶) = (Id‘𝐶)
8723, 81, 82, 86, 84subcid 17109 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘(𝐶cat 𝐻))) → ((Id‘𝐶)‘𝑥) = ((Id‘(𝐶cat 𝐻))‘𝑥))
8887eqcomd 2804 . . . . . 6 ((𝜑𝑥 ∈ (Base‘(𝐶cat 𝐻))) → ((Id‘(𝐶cat 𝐻))‘𝑥) = ((Id‘𝐶)‘𝑥))
8985, 88fveq12d 6652 . . . . 5 ((𝜑𝑥 ∈ (Base‘(𝐶cat 𝐻))) → ((𝑥(2nd ‘(𝐹f 𝐻))𝑥)‘((Id‘(𝐶cat 𝐻))‘𝑥)) = (((𝑥(2nd𝐹)𝑥) ↾ (𝑥𝐻𝑥))‘((Id‘𝐶)‘𝑥)))
9031adantr 484 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘(𝐶cat 𝐻))) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
9138, 35eqsstrrd 3954 . . . . . . . 8 (𝜑 → (Base‘(𝐶cat 𝐻)) ⊆ (Base‘𝐶))
9291sselda 3915 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘(𝐶cat 𝐻))) → 𝑥 ∈ (Base‘𝐶))
9328, 86, 20, 90, 92funcid 17132 . . . . . 6 ((𝜑𝑥 ∈ (Base‘(𝐶cat 𝐻))) → ((𝑥(2nd𝐹)𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐷)‘((1st𝐹)‘𝑥)))
9481, 82, 84, 86subcidcl 17106 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘(𝐶cat 𝐻))) → ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥))
9594fvresd 6665 . . . . . 6 ((𝜑𝑥 ∈ (Base‘(𝐶cat 𝐻))) → (((𝑥(2nd𝐹)𝑥) ↾ (𝑥𝐻𝑥))‘((Id‘𝐶)‘𝑥)) = ((𝑥(2nd𝐹)𝑥)‘((Id‘𝐶)‘𝑥)))
9684fvresd 6665 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘(𝐶cat 𝐻))) → (((1st𝐹) ↾ dom dom 𝐻)‘𝑥) = ((1st𝐹)‘𝑥))
9796fveq2d 6649 . . . . . 6 ((𝜑𝑥 ∈ (Base‘(𝐶cat 𝐻))) → ((Id‘𝐷)‘(((1st𝐹) ↾ dom dom 𝐻)‘𝑥)) = ((Id‘𝐷)‘((1st𝐹)‘𝑥)))
9893, 95, 973eqtr4d 2843 . . . . 5 ((𝜑𝑥 ∈ (Base‘(𝐶cat 𝐻))) → (((𝑥(2nd𝐹)𝑥) ↾ (𝑥𝐻𝑥))‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐷)‘(((1st𝐹) ↾ dom dom 𝐻)‘𝑥)))
9989, 98eqtrd 2833 . . . 4 ((𝜑𝑥 ∈ (Base‘(𝐶cat 𝐻))) → ((𝑥(2nd ‘(𝐹f 𝐻))𝑥)‘((Id‘(𝐶cat 𝐻))‘𝑥)) = ((Id‘𝐷)‘(((1st𝐹) ↾ dom dom 𝐻)‘𝑥)))
10023ad2ant1 1130 . . . . . . . 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 1130 . . . . . . . 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 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))) → 𝑥 ∈ (Base‘(𝐶cat 𝐻)))
103383ad2ant1 1130 . . . . . . . . 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 2893 . . . . . . . 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 2798 . . . . . . . 8 (comp‘𝐶) = (comp‘𝐶)
106 simp22 1204 . . . . . . . . 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 2893 . . . . . . . 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 1205 . . . . . . . . 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 2893 . . . . . . . 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 1198 . . . . . . . . 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 1130 . . . . . . . . . 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 7152 . . . . . . . . 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 2893 . . . . . . . 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 1199 . . . . . . . . 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 7152 . . . . . . . . 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 2893 . . . . . . . 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 17107 . . . . . . 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 6665 . . . . . 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 1130 . . . . . . 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 1130 . . . . . . . 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 3916 . . . . . . 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 3916 . . . . . . 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 3916 . . . . . . 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 17105 . . . . . . . 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 3916 . . . . . . 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 17105 . . . . . . . 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 3916 . . . . . . 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 17133 . . . . . 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 2833 . . . . 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 1130 . . . . . . 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 17157 . . . . . 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 17094 . . . . . . . . . 10 (𝜑 → (comp‘𝐶) = (comp‘(𝐶cat 𝐻)))
1331323ad2ant1 1130 . . . . . . . . 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 2804 . . . . . . . 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 7152 . . . . . . 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 7152 . . . . . 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 6652 . . . . 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 6665 . . . . . . . 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 6665 . . . . . . . 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 4773 . . . . . . 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 6665 . . . . . . 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 7153 . . . . . 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 17157 . . . . . . . 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 6647 . . . . . . 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 6665 . . . . . . 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 2833 . . . . . 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 17157 . . . . . . . 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 6647 . . . . . . 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 6665 . . . . . . 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 2833 . . . . . 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 7156 . . . . 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 2843 . . . 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 17127 . . 3 (𝜑 → ((1st𝐹) ↾ dom dom 𝐻)((𝐶cat 𝐻) Func 𝐷)(2nd ‘(𝐹f 𝐻)))
154 df-br 5031 . . 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 221 . 2 (𝜑 → ⟨((1st𝐹) ↾ dom dom 𝐻), (2nd ‘(𝐹f 𝐻))⟩ ∈ ((𝐶cat 𝐻) Func 𝐷))
15614, 155eqeltrd 2890 1 (𝜑 → (𝐹f 𝐻) ∈ ((𝐶cat 𝐻) Func 𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  Vcvv 3441   ⊆ wss 3881  ⟨cop 4531   class class class wbr 5030   ↦ cmpt 5110   × cxp 5517  dom cdm 5519   ↾ cres 5521  Rel wrel 5524   Fn wfn 6319  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135  1st c1st 7669  2nd c2nd 7670  Basecbs 16475  Hom chom 16568  compcco 16569  Catccat 16927  Idccid 16928   ↾cat cresc 17070  Subcatcsubc 17071   Func cfunc 17116   ↾f cresf 17119 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-er 8272  df-map 8391  df-pm 8392  df-ixp 8445  df-en 8493  df-dom 8494  df-sdom 8495  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-hom 16581  df-cco 16582  df-cat 16931  df-cid 16932  df-homf 16933  df-ssc 17072  df-resc 17073  df-subc 17074  df-func 17120  df-resf 17123 This theorem is referenced by:  funcrngcsetc  44620  funcringcsetc  44657
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