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Theorem catcfuccl 16980
Description: The category of categories for a weak universe is closed under the functor category operation. (Contributed by Mario Carneiro, 12-Jan-2017.)
Hypotheses
Ref Expression
catcfuccl.c 𝐶 = (CatCat‘𝑈)
catcfuccl.b 𝐵 = (Base‘𝐶)
catcfuccl.o 𝑄 = (𝑋 FuncCat 𝑌)
catcfuccl.u (𝜑𝑈 ∈ WUni)
catcfuccl.1 (𝜑 → ω ∈ 𝑈)
catcfuccl.x (𝜑𝑋𝐵)
catcfuccl.y (𝜑𝑌𝐵)
Assertion
Ref Expression
catcfuccl (𝜑𝑄𝐵)

Proof of Theorem catcfuccl
Dummy variables 𝑎 𝑏 𝑓 𝑔 𝑣 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 catcfuccl.o . . . . 5 𝑄 = (𝑋 FuncCat 𝑌)
2 eqid 2760 . . . . 5 (𝑋 Func 𝑌) = (𝑋 Func 𝑌)
3 eqid 2760 . . . . 5 (𝑋 Nat 𝑌) = (𝑋 Nat 𝑌)
4 eqid 2760 . . . . 5 (Base‘𝑋) = (Base‘𝑋)
5 eqid 2760 . . . . 5 (comp‘𝑌) = (comp‘𝑌)
6 inss2 3977 . . . . . 6 (𝑈 ∩ Cat) ⊆ Cat
7 catcfuccl.x . . . . . . 7 (𝜑𝑋𝐵)
8 catcfuccl.c . . . . . . . 8 𝐶 = (CatCat‘𝑈)
9 catcfuccl.b . . . . . . . 8 𝐵 = (Base‘𝐶)
10 catcfuccl.u . . . . . . . 8 (𝜑𝑈 ∈ WUni)
118, 9, 10catcbas 16968 . . . . . . 7 (𝜑𝐵 = (𝑈 ∩ Cat))
127, 11eleqtrd 2841 . . . . . 6 (𝜑𝑋 ∈ (𝑈 ∩ Cat))
136, 12sseldi 3742 . . . . 5 (𝜑𝑋 ∈ Cat)
14 catcfuccl.y . . . . . . 7 (𝜑𝑌𝐵)
1514, 11eleqtrd 2841 . . . . . 6 (𝜑𝑌 ∈ (𝑈 ∩ Cat))
166, 15sseldi 3742 . . . . 5 (𝜑𝑌 ∈ Cat)
17 eqidd 2761 . . . . 5 (𝜑 → (𝑣 ∈ ((𝑋 Func 𝑌) × (𝑋 Func 𝑌)), ∈ (𝑋 Func 𝑌) ↦ (1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥))))) = (𝑣 ∈ ((𝑋 Func 𝑌) × (𝑋 Func 𝑌)), ∈ (𝑋 Func 𝑌) ↦ (1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥))))))
181, 2, 3, 4, 5, 13, 16, 17fucval 16839 . . . 4 (𝜑𝑄 = {⟨(Base‘ndx), (𝑋 Func 𝑌)⟩, ⟨(Hom ‘ndx), (𝑋 Nat 𝑌)⟩, ⟨(comp‘ndx), (𝑣 ∈ ((𝑋 Func 𝑌) × (𝑋 Func 𝑌)), ∈ (𝑋 Func 𝑌) ↦ (1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥)))))⟩})
19 df-base 16085 . . . . . . 7 Base = Slot 1
20 catcfuccl.1 . . . . . . . 8 (𝜑 → ω ∈ 𝑈)
2110, 20wunndx 16100 . . . . . . 7 (𝜑 → ndx ∈ 𝑈)
2219, 10, 21wunstr 16103 . . . . . 6 (𝜑 → (Base‘ndx) ∈ 𝑈)
23 inss1 3976 . . . . . . . 8 (𝑈 ∩ Cat) ⊆ 𝑈
2423, 12sseldi 3742 . . . . . . 7 (𝜑𝑋𝑈)
2523, 15sseldi 3742 . . . . . . 7 (𝜑𝑌𝑈)
2610, 24, 25wunfunc 16780 . . . . . 6 (𝜑 → (𝑋 Func 𝑌) ∈ 𝑈)
2710, 22, 26wunop 9756 . . . . 5 (𝜑 → ⟨(Base‘ndx), (𝑋 Func 𝑌)⟩ ∈ 𝑈)
28 df-hom 16188 . . . . . . 7 Hom = Slot 14
2928, 10, 21wunstr 16103 . . . . . 6 (𝜑 → (Hom ‘ndx) ∈ 𝑈)
3010, 24, 25wunnat 16837 . . . . . 6 (𝜑 → (𝑋 Nat 𝑌) ∈ 𝑈)
3110, 29, 30wunop 9756 . . . . 5 (𝜑 → ⟨(Hom ‘ndx), (𝑋 Nat 𝑌)⟩ ∈ 𝑈)
32 df-cco 16189 . . . . . . 7 comp = Slot 15
3332, 10, 21wunstr 16103 . . . . . 6 (𝜑 → (comp‘ndx) ∈ 𝑈)
3410, 26, 26wunxp 9758 . . . . . . . 8 (𝜑 → ((𝑋 Func 𝑌) × (𝑋 Func 𝑌)) ∈ 𝑈)
3510, 34, 26wunxp 9758 . . . . . . 7 (𝜑 → (((𝑋 Func 𝑌) × (𝑋 Func 𝑌)) × (𝑋 Func 𝑌)) ∈ 𝑈)
3632, 10, 25wunstr 16103 . . . . . . . . . . . . . 14 (𝜑 → (comp‘𝑌) ∈ 𝑈)
3710, 36wunrn 9763 . . . . . . . . . . . . 13 (𝜑 → ran (comp‘𝑌) ∈ 𝑈)
3810, 37wununi 9740 . . . . . . . . . . . 12 (𝜑 ran (comp‘𝑌) ∈ 𝑈)
3910, 38wunrn 9763 . . . . . . . . . . 11 (𝜑 → ran ran (comp‘𝑌) ∈ 𝑈)
4010, 39wununi 9740 . . . . . . . . . 10 (𝜑 ran ran (comp‘𝑌) ∈ 𝑈)
4110, 40wunpw 9741 . . . . . . . . 9 (𝜑 → 𝒫 ran ran (comp‘𝑌) ∈ 𝑈)
4219, 10, 24wunstr 16103 . . . . . . . . 9 (𝜑 → (Base‘𝑋) ∈ 𝑈)
4310, 41, 42wunmap 9760 . . . . . . . 8 (𝜑 → (𝒫 ran ran (comp‘𝑌) ↑𝑚 (Base‘𝑋)) ∈ 𝑈)
4410, 30wunrn 9763 . . . . . . . . . 10 (𝜑 → ran (𝑋 Nat 𝑌) ∈ 𝑈)
4510, 44wununi 9740 . . . . . . . . 9 (𝜑 ran (𝑋 Nat 𝑌) ∈ 𝑈)
4610, 45, 45wunxp 9758 . . . . . . . 8 (𝜑 → ( ran (𝑋 Nat 𝑌) × ran (𝑋 Nat 𝑌)) ∈ 𝑈)
4710, 43, 46wunpm 9759 . . . . . . 7 (𝜑 → ((𝒫 ran ran (comp‘𝑌) ↑𝑚 (Base‘𝑋)) ↑pm ( ran (𝑋 Nat 𝑌) × ran (𝑋 Nat 𝑌))) ∈ 𝑈)
48 fvex 6363 . . . . . . . . . . 11 (1st𝑣) ∈ V
49 fvex 6363 . . . . . . . . . . . . . 14 (2nd𝑣) ∈ V
50 ovex 6842 . . . . . . . . . . . . . . . . 17 (𝒫 ran ran (comp‘𝑌) ↑𝑚 (Base‘𝑋)) ∈ V
51 ovex 6842 . . . . . . . . . . . . . . . . . . . 20 (𝑋 Nat 𝑌) ∈ V
5251rnex 7266 . . . . . . . . . . . . . . . . . . 19 ran (𝑋 Nat 𝑌) ∈ V
5352uniex 7119 . . . . . . . . . . . . . . . . . 18 ran (𝑋 Nat 𝑌) ∈ V
5453, 53xpex 7128 . . . . . . . . . . . . . . . . 17 ( ran (𝑋 Nat 𝑌) × ran (𝑋 Nat 𝑌)) ∈ V
55 eqid 2760 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥))) = (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥)))
56 ovssunirn 6845 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥)) ⊆ ran (⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))
57 ovssunirn 6845 . . . . . . . . . . . . . . . . . . . . . . . . 25 (⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥)) ⊆ ran (comp‘𝑌)
58 rnss 5509 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥)) ⊆ ran (comp‘𝑌) → ran (⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥)) ⊆ ran ran (comp‘𝑌))
59 uniss 4610 . . . . . . . . . . . . . . . . . . . . . . . . 25 (ran (⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥)) ⊆ ran ran (comp‘𝑌) → ran (⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥)) ⊆ ran ran (comp‘𝑌))
6057, 58, 59mp2b 10 . . . . . . . . . . . . . . . . . . . . . . . 24 ran (⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥)) ⊆ ran ran (comp‘𝑌)
6156, 60sstri 3753 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥)) ⊆ ran ran (comp‘𝑌)
62 ovex 6842 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥)) ∈ V
6362elpw 4308 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥)) ∈ 𝒫 ran ran (comp‘𝑌) ↔ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥)) ⊆ ran ran (comp‘𝑌))
6461, 63mpbir 221 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥)) ∈ 𝒫 ran ran (comp‘𝑌)
6564a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (Base‘𝑋) → ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥)) ∈ 𝒫 ran ran (comp‘𝑌))
6655, 65fmpti 6547 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥))):(Base‘𝑋)⟶𝒫 ran ran (comp‘𝑌)
67 fvex 6363 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (comp‘𝑌) ∈ V
6867rnex 7266 . . . . . . . . . . . . . . . . . . . . . . . . 25 ran (comp‘𝑌) ∈ V
6968uniex 7119 . . . . . . . . . . . . . . . . . . . . . . . 24 ran (comp‘𝑌) ∈ V
7069rnex 7266 . . . . . . . . . . . . . . . . . . . . . . 23 ran ran (comp‘𝑌) ∈ V
7170uniex 7119 . . . . . . . . . . . . . . . . . . . . . 22 ran ran (comp‘𝑌) ∈ V
7271pwex 4997 . . . . . . . . . . . . . . . . . . . . 21 𝒫 ran ran (comp‘𝑌) ∈ V
73 fvex 6363 . . . . . . . . . . . . . . . . . . . . 21 (Base‘𝑋) ∈ V
7472, 73elmap 8054 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 ∈ (Base‘𝑋) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥))) ∈ (𝒫 ran ran (comp‘𝑌) ↑𝑚 (Base‘𝑋)) ↔ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥))):(Base‘𝑋)⟶𝒫 ran ran (comp‘𝑌))
7566, 74mpbir 221 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥))) ∈ (𝒫 ran ran (comp‘𝑌) ↑𝑚 (Base‘𝑋))
7675rgen2w 3063 . . . . . . . . . . . . . . . . . 18 𝑏 ∈ (𝑔(𝑋 Nat 𝑌))∀𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔)(𝑥 ∈ (Base‘𝑋) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥))) ∈ (𝒫 ran ran (comp‘𝑌) ↑𝑚 (Base‘𝑋))
77 eqid 2760 . . . . . . . . . . . . . . . . . . 19 (𝑏 ∈ (𝑔(𝑋 Nat 𝑌)), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥)))) = (𝑏 ∈ (𝑔(𝑋 Nat 𝑌)), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥))))
7877fmpt2 7406 . . . . . . . . . . . . . . . . . 18 (∀𝑏 ∈ (𝑔(𝑋 Nat 𝑌))∀𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔)(𝑥 ∈ (Base‘𝑋) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥))) ∈ (𝒫 ran ran (comp‘𝑌) ↑𝑚 (Base‘𝑋)) ↔ (𝑏 ∈ (𝑔(𝑋 Nat 𝑌)), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥)))):((𝑔(𝑋 Nat 𝑌)) × (𝑓(𝑋 Nat 𝑌)𝑔))⟶(𝒫 ran ran (comp‘𝑌) ↑𝑚 (Base‘𝑋)))
7976, 78mpbi 220 . . . . . . . . . . . . . . . . 17 (𝑏 ∈ (𝑔(𝑋 Nat 𝑌)), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥)))):((𝑔(𝑋 Nat 𝑌)) × (𝑓(𝑋 Nat 𝑌)𝑔))⟶(𝒫 ran ran (comp‘𝑌) ↑𝑚 (Base‘𝑋))
80 ovssunirn 6845 . . . . . . . . . . . . . . . . . 18 (𝑔(𝑋 Nat 𝑌)) ⊆ ran (𝑋 Nat 𝑌)
81 ovssunirn 6845 . . . . . . . . . . . . . . . . . 18 (𝑓(𝑋 Nat 𝑌)𝑔) ⊆ ran (𝑋 Nat 𝑌)
82 xpss12 5281 . . . . . . . . . . . . . . . . . 18 (((𝑔(𝑋 Nat 𝑌)) ⊆ ran (𝑋 Nat 𝑌) ∧ (𝑓(𝑋 Nat 𝑌)𝑔) ⊆ ran (𝑋 Nat 𝑌)) → ((𝑔(𝑋 Nat 𝑌)) × (𝑓(𝑋 Nat 𝑌)𝑔)) ⊆ ( ran (𝑋 Nat 𝑌) × ran (𝑋 Nat 𝑌)))
8380, 81, 82mp2an 710 . . . . . . . . . . . . . . . . 17 ((𝑔(𝑋 Nat 𝑌)) × (𝑓(𝑋 Nat 𝑌)𝑔)) ⊆ ( ran (𝑋 Nat 𝑌) × ran (𝑋 Nat 𝑌))
84 elpm2r 8043 . . . . . . . . . . . . . . . . 17 ((((𝒫 ran ran (comp‘𝑌) ↑𝑚 (Base‘𝑋)) ∈ V ∧ ( ran (𝑋 Nat 𝑌) × ran (𝑋 Nat 𝑌)) ∈ V) ∧ ((𝑏 ∈ (𝑔(𝑋 Nat 𝑌)), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥)))):((𝑔(𝑋 Nat 𝑌)) × (𝑓(𝑋 Nat 𝑌)𝑔))⟶(𝒫 ran ran (comp‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ ((𝑔(𝑋 Nat 𝑌)) × (𝑓(𝑋 Nat 𝑌)𝑔)) ⊆ ( ran (𝑋 Nat 𝑌) × ran (𝑋 Nat 𝑌)))) → (𝑏 ∈ (𝑔(𝑋 Nat 𝑌)), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥)))) ∈ ((𝒫 ran ran (comp‘𝑌) ↑𝑚 (Base‘𝑋)) ↑pm ( ran (𝑋 Nat 𝑌) × ran (𝑋 Nat 𝑌))))
8550, 54, 79, 83, 84mp4an 711 . . . . . . . . . . . . . . . 16 (𝑏 ∈ (𝑔(𝑋 Nat 𝑌)), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥)))) ∈ ((𝒫 ran ran (comp‘𝑌) ↑𝑚 (Base‘𝑋)) ↑pm ( ran (𝑋 Nat 𝑌) × ran (𝑋 Nat 𝑌)))
8685sbcth 3591 . . . . . . . . . . . . . . 15 ((2nd𝑣) ∈ V → [(2nd𝑣) / 𝑔](𝑏 ∈ (𝑔(𝑋 Nat 𝑌)), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥)))) ∈ ((𝒫 ran ran (comp‘𝑌) ↑𝑚 (Base‘𝑋)) ↑pm ( ran (𝑋 Nat 𝑌) × ran (𝑋 Nat 𝑌))))
87 sbcel1g 4130 . . . . . . . . . . . . . . 15 ((2nd𝑣) ∈ V → ([(2nd𝑣) / 𝑔](𝑏 ∈ (𝑔(𝑋 Nat 𝑌)), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥)))) ∈ ((𝒫 ran ran (comp‘𝑌) ↑𝑚 (Base‘𝑋)) ↑pm ( ran (𝑋 Nat 𝑌) × ran (𝑋 Nat 𝑌))) ↔ (2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥)))) ∈ ((𝒫 ran ran (comp‘𝑌) ↑𝑚 (Base‘𝑋)) ↑pm ( ran (𝑋 Nat 𝑌) × ran (𝑋 Nat 𝑌)))))
8886, 87mpbid 222 . . . . . . . . . . . . . 14 ((2nd𝑣) ∈ V → (2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥)))) ∈ ((𝒫 ran ran (comp‘𝑌) ↑𝑚 (Base‘𝑋)) ↑pm ( ran (𝑋 Nat 𝑌) × ran (𝑋 Nat 𝑌))))
8949, 88ax-mp 5 . . . . . . . . . . . . 13 (2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥)))) ∈ ((𝒫 ran ran (comp‘𝑌) ↑𝑚 (Base‘𝑋)) ↑pm ( ran (𝑋 Nat 𝑌) × ran (𝑋 Nat 𝑌)))
9089sbcth 3591 . . . . . . . . . . . 12 ((1st𝑣) ∈ V → [(1st𝑣) / 𝑓](2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥)))) ∈ ((𝒫 ran ran (comp‘𝑌) ↑𝑚 (Base‘𝑋)) ↑pm ( ran (𝑋 Nat 𝑌) × ran (𝑋 Nat 𝑌))))
91 sbcel1g 4130 . . . . . . . . . . . 12 ((1st𝑣) ∈ V → ([(1st𝑣) / 𝑓](2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥)))) ∈ ((𝒫 ran ran (comp‘𝑌) ↑𝑚 (Base‘𝑋)) ↑pm ( ran (𝑋 Nat 𝑌) × ran (𝑋 Nat 𝑌))) ↔ (1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥)))) ∈ ((𝒫 ran ran (comp‘𝑌) ↑𝑚 (Base‘𝑋)) ↑pm ( ran (𝑋 Nat 𝑌) × ran (𝑋 Nat 𝑌)))))
9290, 91mpbid 222 . . . . . . . . . . 11 ((1st𝑣) ∈ V → (1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥)))) ∈ ((𝒫 ran ran (comp‘𝑌) ↑𝑚 (Base‘𝑋)) ↑pm ( ran (𝑋 Nat 𝑌) × ran (𝑋 Nat 𝑌))))
9348, 92ax-mp 5 . . . . . . . . . 10 (1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥)))) ∈ ((𝒫 ran ran (comp‘𝑌) ↑𝑚 (Base‘𝑋)) ↑pm ( ran (𝑋 Nat 𝑌) × ran (𝑋 Nat 𝑌)))
9493rgen2w 3063 . . . . . . . . 9 𝑣 ∈ ((𝑋 Func 𝑌) × (𝑋 Func 𝑌))∀ ∈ (𝑋 Func 𝑌)(1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥)))) ∈ ((𝒫 ran ran (comp‘𝑌) ↑𝑚 (Base‘𝑋)) ↑pm ( ran (𝑋 Nat 𝑌) × ran (𝑋 Nat 𝑌)))
95 eqid 2760 . . . . . . . . . 10 (𝑣 ∈ ((𝑋 Func 𝑌) × (𝑋 Func 𝑌)), ∈ (𝑋 Func 𝑌) ↦ (1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥))))) = (𝑣 ∈ ((𝑋 Func 𝑌) × (𝑋 Func 𝑌)), ∈ (𝑋 Func 𝑌) ↦ (1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥)))))
9695fmpt2 7406 . . . . . . . . 9 (∀𝑣 ∈ ((𝑋 Func 𝑌) × (𝑋 Func 𝑌))∀ ∈ (𝑋 Func 𝑌)(1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥)))) ∈ ((𝒫 ran ran (comp‘𝑌) ↑𝑚 (Base‘𝑋)) ↑pm ( ran (𝑋 Nat 𝑌) × ran (𝑋 Nat 𝑌))) ↔ (𝑣 ∈ ((𝑋 Func 𝑌) × (𝑋 Func 𝑌)), ∈ (𝑋 Func 𝑌) ↦ (1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥))))):(((𝑋 Func 𝑌) × (𝑋 Func 𝑌)) × (𝑋 Func 𝑌))⟶((𝒫 ran ran (comp‘𝑌) ↑𝑚 (Base‘𝑋)) ↑pm ( ran (𝑋 Nat 𝑌) × ran (𝑋 Nat 𝑌))))
9794, 96mpbi 220 . . . . . . . 8 (𝑣 ∈ ((𝑋 Func 𝑌) × (𝑋 Func 𝑌)), ∈ (𝑋 Func 𝑌) ↦ (1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥))))):(((𝑋 Func 𝑌) × (𝑋 Func 𝑌)) × (𝑋 Func 𝑌))⟶((𝒫 ran ran (comp‘𝑌) ↑𝑚 (Base‘𝑋)) ↑pm ( ran (𝑋 Nat 𝑌) × ran (𝑋 Nat 𝑌)))
9897a1i 11 . . . . . . 7 (𝜑 → (𝑣 ∈ ((𝑋 Func 𝑌) × (𝑋 Func 𝑌)), ∈ (𝑋 Func 𝑌) ↦ (1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥))))):(((𝑋 Func 𝑌) × (𝑋 Func 𝑌)) × (𝑋 Func 𝑌))⟶((𝒫 ran ran (comp‘𝑌) ↑𝑚 (Base‘𝑋)) ↑pm ( ran (𝑋 Nat 𝑌) × ran (𝑋 Nat 𝑌))))
9910, 35, 47, 98wunf 9761 . . . . . 6 (𝜑 → (𝑣 ∈ ((𝑋 Func 𝑌) × (𝑋 Func 𝑌)), ∈ (𝑋 Func 𝑌) ↦ (1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥))))) ∈ 𝑈)
10010, 33, 99wunop 9756 . . . . 5 (𝜑 → ⟨(comp‘ndx), (𝑣 ∈ ((𝑋 Func 𝑌) × (𝑋 Func 𝑌)), ∈ (𝑋 Func 𝑌) ↦ (1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥)))))⟩ ∈ 𝑈)
10110, 27, 31, 100wuntp 9745 . . . 4 (𝜑 → {⟨(Base‘ndx), (𝑋 Func 𝑌)⟩, ⟨(Hom ‘ndx), (𝑋 Nat 𝑌)⟩, ⟨(comp‘ndx), (𝑣 ∈ ((𝑋 Func 𝑌) × (𝑋 Func 𝑌)), ∈ (𝑋 Func 𝑌) ↦ (1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥)))))⟩} ∈ 𝑈)
10218, 101eqeltrd 2839 . . 3 (𝜑𝑄𝑈)
1031, 13, 16fuccat 16851 . . 3 (𝜑𝑄 ∈ Cat)
104102, 103elind 3941 . 2 (𝜑𝑄 ∈ (𝑈 ∩ Cat))
105104, 11eleqtrrd 2842 1 (𝜑𝑄𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  wral 3050  Vcvv 3340  [wsbc 3576  csb 3674  cin 3714  wss 3715  𝒫 cpw 4302  {ctp 4325  cop 4327   cuni 4588  cmpt 4881   × cxp 5264  ran crn 5267  wf 6045  cfv 6049  (class class class)co 6814  cmpt2 6816  ωcom 7231  1st c1st 7332  2nd c2nd 7333  𝑚 cmap 8025  pm cpm 8026  WUnicwun 9734  1c1 10149  4c4 11284  5c5 11285  cdc 11705  ndxcnx 16076  Basecbs 16079  Hom chom 16174  compcco 16175  Catccat 16546   Func cfunc 16735   Nat cnat 16822   FuncCat cfuc 16823  CatCatccatc 16965
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 7115  ax-inf2 8713  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  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-nel 3036  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-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  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-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  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 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-omul 7735  df-er 7913  df-ec 7915  df-qs 7919  df-map 8027  df-pm 8028  df-ixp 8077  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-wun 9736  df-ni 9906  df-pli 9907  df-mi 9908  df-lti 9909  df-plpq 9942  df-mpq 9943  df-ltpq 9944  df-enq 9945  df-nq 9946  df-erq 9947  df-plq 9948  df-mq 9949  df-1nq 9950  df-rq 9951  df-ltnq 9952  df-np 10015  df-plp 10017  df-ltp 10019  df-enr 10089  df-nr 10090  df-c 10154  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-nn 11233  df-2 11291  df-3 11292  df-4 11293  df-5 11294  df-6 11295  df-7 11296  df-8 11297  df-9 11298  df-n0 11505  df-z 11590  df-dec 11706  df-uz 11900  df-fz 12540  df-struct 16081  df-ndx 16082  df-slot 16083  df-base 16085  df-hom 16188  df-cco 16189  df-cat 16550  df-cid 16551  df-func 16739  df-nat 16824  df-fuc 16825  df-catc 16966
This theorem is referenced by: (None)
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