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Theorem funcres 17946
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 17942 . . 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 6048 . . . . . 6 ((1st𝐹) ↾ dom dom 𝐻) ∈ V
7 dmexg 7923 . . . . . . 7 (𝐻 ∈ (Subcat‘𝐶) → dom 𝐻 ∈ V)
8 mptexg 7240 . . . . . . 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 8025 . . . . . 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 2774 . . . 4 (𝜑 → (2nd ‘(𝐹f 𝐻)) = (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧))))
1312opeq2d 4884 . . 3 (𝜑 → ⟨((1st𝐹) ↾ dom dom 𝐻), (2nd ‘(𝐹f 𝐻))⟩ = ⟨((1st𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)))⟩)
143, 13eqtr4d 2777 . 2 (𝜑 → (𝐹f 𝐻) = ⟨((1st𝐹) ↾ dom dom 𝐻), (2nd ‘(𝐹f 𝐻))⟩)
15 eqid 2734 . . . 4 (Base‘(𝐶cat 𝐻)) = (Base‘(𝐶cat 𝐻))
16 eqid 2734 . . . 4 (Base‘𝐷) = (Base‘𝐷)
17 eqid 2734 . . . 4 (Hom ‘(𝐶cat 𝐻)) = (Hom ‘(𝐶cat 𝐻))
18 eqid 2734 . . . 4 (Hom ‘𝐷) = (Hom ‘𝐷)
19 eqid 2734 . . . 4 (Id‘(𝐶cat 𝐻)) = (Id‘(𝐶cat 𝐻))
20 eqid 2734 . . . 4 (Id‘𝐷) = (Id‘𝐷)
21 eqid 2734 . . . 4 (comp‘(𝐶cat 𝐻)) = (comp‘(𝐶cat 𝐻))
22 eqid 2734 . . . 4 (comp‘𝐷) = (comp‘𝐷)
23 eqid 2734 . . . . 5 (𝐶cat 𝐻) = (𝐶cat 𝐻)
2423, 2subccat 17898 . . . 4 (𝜑 → (𝐶cat 𝐻) ∈ Cat)
25 funcrcl 17913 . . . . . 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 2734 . . . . . . 7 (Base‘𝐶) = (Base‘𝐶)
29 relfunc 17912 . . . . . . . 8 Rel (𝐶 Func 𝐷)
30 1st2ndbr 8065 . . . . . . . 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 17916 . . . . . 6 (𝜑 → (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷))
33 eqidd 2735 . . . . . . . 8 (𝜑 → dom dom 𝐻 = dom dom 𝐻)
342, 33subcfn 17891 . . . . . . 7 (𝜑𝐻 Fn (dom dom 𝐻 × dom dom 𝐻))
352, 34, 28subcss1 17892 . . . . . 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 17876 . . . . . 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 6048 . . . . . 6 (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)) ∈ V
43 eqid 2734 . . . . . 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 2740 . . . . . 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 5720 . . . . . . 7 (𝜑 → (dom dom 𝐻 × dom dom 𝐻) = ((Base‘(𝐶cat 𝐻)) × (Base‘(𝐶cat 𝐻))))
4947, 48eqtrd 2774 . . . . . 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 2734 . . . . . . . 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 2841 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → 𝑥 ∈ dom dom 𝐻)
5854, 57sseldd 3995 . . . . . . . 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 2841 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → 𝑦 ∈ dom dom 𝐻)
6154, 60sseldd 3995 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → 𝑦 ∈ (Base‘𝐶))
6228, 52, 18, 53, 58, 61funcf2 17918 . . . . . . 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 17893 . . . . . . 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 17945 . . . . . . 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 17878 . . . . . . . 8 (𝜑𝐻 = (Hom ‘(𝐶cat 𝐻)))
7271adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶cat 𝐻)))) → 𝐻 = (Hom ‘(𝐶cat 𝐻)))
7372oveqd 7447 . . . . . 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 7448 . . . . . . 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 2740 . . . . . 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 2824 . . . . . . . 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 17945 . . . . . 6 ((𝜑𝑥 ∈ (Base‘(𝐶cat 𝐻))) → (𝑥(2nd ‘(𝐹f 𝐻))𝑥) = ((𝑥(2nd𝐹)𝑥) ↾ (𝑥𝐻𝑥)))
86 eqid 2734 . . . . . . . 8 (Id‘𝐶) = (Id‘𝐶)
8723, 81, 82, 86, 84subcid 17897 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘(𝐶cat 𝐻))) → ((Id‘𝐶)‘𝑥) = ((Id‘(𝐶cat 𝐻))‘𝑥))
8887eqcomd 2740 . . . . . 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 4034 . . . . . . . 8 (𝜑 → (Base‘(𝐶cat 𝐻)) ⊆ (Base‘𝐶))
9291sselda 3994 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘(𝐶cat 𝐻))) → 𝑥 ∈ (Base‘𝐶))
9328, 86, 20, 90, 92funcid 17920 . . . . . 6 ((𝜑𝑥 ∈ (Base‘(𝐶cat 𝐻))) → ((𝑥(2nd𝐹)𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐷)‘((1st𝐹)‘𝑥)))
9481, 82, 84, 86subcidcl 17894 . . . . . . 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 2784 . . . . 5 ((𝜑𝑥 ∈ (Base‘(𝐶cat 𝐻))) → (((𝑥(2nd𝐹)𝑥) ↾ (𝑥𝐻𝑥))‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐷)‘(((1st𝐹) ↾ dom dom 𝐻)‘𝑥)))
9989, 98eqtrd 2774 . . . 4 ((𝜑𝑥 ∈ (Base‘(𝐶cat 𝐻))) → ((𝑥(2nd ‘(𝐹f 𝐻))𝑥)‘((Id‘(𝐶cat 𝐻))‘𝑥)) = ((Id‘𝐷)‘(((1st𝐹) ↾ dom dom 𝐻)‘𝑥)))
10023ad2ant1 1132 . . . . . . . 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 1132 . . . . . . . 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 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))) → 𝑥 ∈ (Base‘(𝐶cat 𝐻)))
103383ad2ant1 1132 . . . . . . . . 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 2841 . . . . . . . 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 2734 . . . . . . . 8 (comp‘𝐶) = (comp‘𝐶)
106 simp22 1206 . . . . . . . . 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 2841 . . . . . . . 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 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))) → 𝑧 ∈ (Base‘(𝐶cat 𝐻)))
109108, 103eleqtrrd 2841 . . . . . . . 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 1200 . . . . . . . . 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 1132 . . . . . . . . . 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 7447 . . . . . . . . 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 2841 . . . . . . . 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 1201 . . . . . . . . 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 7447 . . . . . . . . 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 2841 . . . . . . . 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 17895 . . . . . . 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 1132 . . . . . . 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 1132 . . . . . . . 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 3995 . . . . . . 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 3995 . . . . . . 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 3995 . . . . . . 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 17893 . . . . . . . 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 3995 . . . . . . 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 17893 . . . . . . . 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 3995 . . . . . . 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 17921 . . . . . 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 2774 . . . . 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 1132 . . . . . . 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 17945 . . . . . 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 17880 . . . . . . . . . 10 (𝜑 → (comp‘𝐶) = (comp‘(𝐶cat 𝐻)))
1331323ad2ant1 1132 . . . . . . . . 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 2740 . . . . . . . 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 7447 . . . . . . 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 7447 . . . . . 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 4885 . . . . . . 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 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))) → (⟨(((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 17945 . . . . . . . 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 2774 . . . . . 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 17945 . . . . . . . 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 2774 . . . . . 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 7451 . . . . 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 2784 . . . 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 17915 . . 3 (𝜑 → ((1st𝐹) ↾ dom dom 𝐻)((𝐶cat 𝐻) Func 𝐷)(2nd ‘(𝐹f 𝐻)))
154 df-br 5148 . . 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 2838 1 (𝜑 → (𝐹f 𝐻) ∈ ((𝐶cat 𝐻) Func 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1536  wcel 2105  Vcvv 3477  wss 3962  cop 4636   class class class wbr 5147  cmpt 5230   × cxp 5686  dom cdm 5688  cres 5690  Rel wrel 5693   Fn wfn 6557  wf 6558  cfv 6562  (class class class)co 7430  1st c1st 8010  2nd c2nd 8011  Basecbs 17244  Hom chom 17308  compcco 17309  Catccat 17708  Idccid 17709  cat cresc 17855  Subcatcsubc 17856   Func cfunc 17904  f cresf 17907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-er 8743  df-map 8866  df-pm 8867  df-ixp 8936  df-en 8984  df-dom 8985  df-sdom 8986  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-dec 12731  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-hom 17321  df-cco 17322  df-cat 17712  df-cid 17713  df-homf 17714  df-ssc 17857  df-resc 17858  df-subc 17859  df-func 17908  df-resf 17911
This theorem is referenced by:  funcrngcsetc  20656  funcringcsetc  20690
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