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Theorem catcfuccl 17073
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 2799 . . . . 5 (𝑋 Func 𝑌) = (𝑋 Func 𝑌)
3 eqid 2799 . . . . 5 (𝑋 Nat 𝑌) = (𝑋 Nat 𝑌)
4 eqid 2799 . . . . 5 (Base‘𝑋) = (Base‘𝑋)
5 eqid 2799 . . . . 5 (comp‘𝑌) = (comp‘𝑌)
6 inss2 4029 . . . . . 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 17061 . . . . . . 7 (𝜑𝐵 = (𝑈 ∩ Cat))
127, 11eleqtrd 2880 . . . . . 6 (𝜑𝑋 ∈ (𝑈 ∩ Cat))
136, 12sseldi 3796 . . . . 5 (𝜑𝑋 ∈ Cat)
14 catcfuccl.y . . . . . . 7 (𝜑𝑌𝐵)
1514, 11eleqtrd 2880 . . . . . 6 (𝜑𝑌 ∈ (𝑈 ∩ Cat))
166, 15sseldi 3796 . . . . 5 (𝜑𝑌 ∈ Cat)
17 eqidd 2800 . . . . 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 16932 . . . 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 16190 . . . . . . 7 Base = Slot 1
20 catcfuccl.1 . . . . . . . 8 (𝜑 → ω ∈ 𝑈)
2110, 20wunndx 16205 . . . . . . 7 (𝜑 → ndx ∈ 𝑈)
2219, 10, 21wunstr 16208 . . . . . 6 (𝜑 → (Base‘ndx) ∈ 𝑈)
23 inss1 4028 . . . . . . . 8 (𝑈 ∩ Cat) ⊆ 𝑈
2423, 12sseldi 3796 . . . . . . 7 (𝜑𝑋𝑈)
2523, 15sseldi 3796 . . . . . . 7 (𝜑𝑌𝑈)
2610, 24, 25wunfunc 16873 . . . . . 6 (𝜑 → (𝑋 Func 𝑌) ∈ 𝑈)
2710, 22, 26wunop 9832 . . . . 5 (𝜑 → ⟨(Base‘ndx), (𝑋 Func 𝑌)⟩ ∈ 𝑈)
28 df-hom 16291 . . . . . . 7 Hom = Slot 14
2928, 10, 21wunstr 16208 . . . . . 6 (𝜑 → (Hom ‘ndx) ∈ 𝑈)
3010, 24, 25wunnat 16930 . . . . . 6 (𝜑 → (𝑋 Nat 𝑌) ∈ 𝑈)
3110, 29, 30wunop 9832 . . . . 5 (𝜑 → ⟨(Hom ‘ndx), (𝑋 Nat 𝑌)⟩ ∈ 𝑈)
32 df-cco 16292 . . . . . . 7 comp = Slot 15
3332, 10, 21wunstr 16208 . . . . . 6 (𝜑 → (comp‘ndx) ∈ 𝑈)
3410, 26, 26wunxp 9834 . . . . . . . 8 (𝜑 → ((𝑋 Func 𝑌) × (𝑋 Func 𝑌)) ∈ 𝑈)
3510, 34, 26wunxp 9834 . . . . . . 7 (𝜑 → (((𝑋 Func 𝑌) × (𝑋 Func 𝑌)) × (𝑋 Func 𝑌)) ∈ 𝑈)
3632, 10, 25wunstr 16208 . . . . . . . . . . . . . 14 (𝜑 → (comp‘𝑌) ∈ 𝑈)
3710, 36wunrn 9839 . . . . . . . . . . . . 13 (𝜑 → ran (comp‘𝑌) ∈ 𝑈)
3810, 37wununi 9816 . . . . . . . . . . . 12 (𝜑 ran (comp‘𝑌) ∈ 𝑈)
3910, 38wunrn 9839 . . . . . . . . . . 11 (𝜑 → ran ran (comp‘𝑌) ∈ 𝑈)
4010, 39wununi 9816 . . . . . . . . . 10 (𝜑 ran ran (comp‘𝑌) ∈ 𝑈)
4110, 40wunpw 9817 . . . . . . . . 9 (𝜑 → 𝒫 ran ran (comp‘𝑌) ∈ 𝑈)
4219, 10, 24wunstr 16208 . . . . . . . . 9 (𝜑 → (Base‘𝑋) ∈ 𝑈)
4310, 41, 42wunmap 9836 . . . . . . . 8 (𝜑 → (𝒫 ran ran (comp‘𝑌) ↑𝑚 (Base‘𝑋)) ∈ 𝑈)
4410, 30wunrn 9839 . . . . . . . . . 10 (𝜑 → ran (𝑋 Nat 𝑌) ∈ 𝑈)
4510, 44wununi 9816 . . . . . . . . 9 (𝜑 ran (𝑋 Nat 𝑌) ∈ 𝑈)
4610, 45, 45wunxp 9834 . . . . . . . 8 (𝜑 → ( ran (𝑋 Nat 𝑌) × ran (𝑋 Nat 𝑌)) ∈ 𝑈)
4710, 43, 46wunpm 9835 . . . . . . 7 (𝜑 → ((𝒫 ran ran (comp‘𝑌) ↑𝑚 (Base‘𝑋)) ↑pm ( ran (𝑋 Nat 𝑌) × ran (𝑋 Nat 𝑌))) ∈ 𝑈)
48 fvex 6424 . . . . . . . . . . 11 (1st𝑣) ∈ V
49 fvex 6424 . . . . . . . . . . . . . 14 (2nd𝑣) ∈ V
50 ovex 6910 . . . . . . . . . . . . . . . . 17 (𝒫 ran ran (comp‘𝑌) ↑𝑚 (Base‘𝑋)) ∈ V
51 ovex 6910 . . . . . . . . . . . . . . . . . . . 20 (𝑋 Nat 𝑌) ∈ V
5251rnex 7335 . . . . . . . . . . . . . . . . . . 19 ran (𝑋 Nat 𝑌) ∈ V
5352uniex 7187 . . . . . . . . . . . . . . . . . 18 ran (𝑋 Nat 𝑌) ∈ V
5453, 53xpex 7196 . . . . . . . . . . . . . . . . 17 ( ran (𝑋 Nat 𝑌) × ran (𝑋 Nat 𝑌)) ∈ V
55 eqid 2799 . . . . . . . . . . . . . . . . . . . . 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 6913 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥)) ⊆ ran (⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))
57 ovssunirn 6913 . . . . . . . . . . . . . . . . . . . . . . . . 25 (⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥)) ⊆ ran (comp‘𝑌)
58 rnss 5557 . . . . . . . . . . . . . . . . . . . . . . . . 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 4651 . . . . . . . . . . . . . . . . . . . . . . . . 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 3807 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥)) ⊆ ran ran (comp‘𝑌)
62 ovex 6910 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥)) ∈ V
6362elpw 4355 . . . . . . . . . . . . . . . . . . . . . . 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 223 . . . . . . . . . . . . . . . . . . . . . 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 6608 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥))):(Base‘𝑋)⟶𝒫 ran ran (comp‘𝑌)
67 fvex 6424 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (comp‘𝑌) ∈ V
6867rnex 7335 . . . . . . . . . . . . . . . . . . . . . . . . 25 ran (comp‘𝑌) ∈ V
6968uniex 7187 . . . . . . . . . . . . . . . . . . . . . . . 24 ran (comp‘𝑌) ∈ V
7069rnex 7335 . . . . . . . . . . . . . . . . . . . . . . 23 ran ran (comp‘𝑌) ∈ V
7170uniex 7187 . . . . . . . . . . . . . . . . . . . . . 22 ran ran (comp‘𝑌) ∈ V
7271pwex 5050 . . . . . . . . . . . . . . . . . . . . 21 𝒫 ran ran (comp‘𝑌) ∈ V
73 fvex 6424 . . . . . . . . . . . . . . . . . . . . 21 (Base‘𝑋) ∈ V
7472, 73elmap 8124 . . . . . . . . . . . . . . . . . . . 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 223 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥))) ∈ (𝒫 ran ran (comp‘𝑌) ↑𝑚 (Base‘𝑋))
7675rgen2w 3106 . . . . . . . . . . . . . . . . . 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 2799 . . . . . . . . . . . . . . . . . . 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 7473 . . . . . . . . . . . . . . . . . 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 222 . . . . . . . . . . . . . . . . 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 6913 . . . . . . . . . . . . . . . . . 18 (𝑔(𝑋 Nat 𝑌)) ⊆ ran (𝑋 Nat 𝑌)
81 ovssunirn 6913 . . . . . . . . . . . . . . . . . 18 (𝑓(𝑋 Nat 𝑌)𝑔) ⊆ ran (𝑋 Nat 𝑌)
82 xpss12 5327 . . . . . . . . . . . . . . . . . 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 684 . . . . . . . . . . . . . . . . 17 ((𝑔(𝑋 Nat 𝑌)) × (𝑓(𝑋 Nat 𝑌)𝑔)) ⊆ ( ran (𝑋 Nat 𝑌) × ran (𝑋 Nat 𝑌))
84 elpm2r 8113 . . . . . . . . . . . . . . . . 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 685 . . . . . . . . . . . . . . . 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 3648 . . . . . . . . . . . . . . 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 4182 . . . . . . . . . . . . . . 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 224 . . . . . . . . . . . . . 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 3648 . . . . . . . . . . . 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 4182 . . . . . . . . . . . 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 224 . . . . . . . . . . 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 3106 . . . . . . . . 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 2799 . . . . . . . . . 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 7473 . . . . . . . . 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 222 . . . . . . . 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 9837 . . . . . 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 9832 . . . . 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 9821 . . . 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 2878 . . 3 (𝜑𝑄𝑈)
1031, 13, 16fuccat 16944 . . 3 (𝜑𝑄 ∈ Cat)
104102, 103elind 3996 . 2 (𝜑𝑄 ∈ (𝑈 ∩ Cat))
105104, 11eleqtrrd 2881 1 (𝜑𝑄𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wcel 2157  wral 3089  Vcvv 3385  [wsbc 3633  csb 3728  cin 3768  wss 3769  𝒫 cpw 4349  {ctp 4372  cop 4374   cuni 4628  cmpt 4922   × cxp 5310  ran crn 5313  wf 6097  cfv 6101  (class class class)co 6878  cmpt2 6880  ωcom 7299  1st c1st 7399  2nd c2nd 7400  𝑚 cmap 8095  pm cpm 8096  WUnicwun 9810  1c1 10225  4c4 11370  5c5 11371  cdc 11783  ndxcnx 16181  Basecbs 16184  Hom chom 16278  compcco 16279  Catccat 16639   Func cfunc 16828   Nat cnat 16915   FuncCat cfuc 16916  CatCatccatc 17058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-inf2 8788  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-fal 1667  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-1st 7401  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-1o 7799  df-oadd 7803  df-omul 7804  df-er 7982  df-ec 7984  df-qs 7988  df-map 8097  df-pm 8098  df-ixp 8149  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-wun 9812  df-ni 9982  df-pli 9983  df-mi 9984  df-lti 9985  df-plpq 10018  df-mpq 10019  df-ltpq 10020  df-enq 10021  df-nq 10022  df-erq 10023  df-plq 10024  df-mq 10025  df-1nq 10026  df-rq 10027  df-ltnq 10028  df-np 10091  df-plp 10093  df-ltp 10095  df-enr 10165  df-nr 10166  df-c 10230  df-pnf 10365  df-mnf 10366  df-xr 10367  df-ltxr 10368  df-le 10369  df-sub 10558  df-neg 10559  df-nn 11313  df-2 11376  df-3 11377  df-4 11378  df-5 11379  df-6 11380  df-7 11381  df-8 11382  df-9 11383  df-n0 11581  df-z 11667  df-dec 11784  df-uz 11931  df-fz 12581  df-struct 16186  df-ndx 16187  df-slot 16188  df-base 16190  df-hom 16291  df-cco 16292  df-cat 16643  df-cid 16644  df-func 16832  df-nat 16917  df-fuc 16918  df-catc 17059
This theorem is referenced by: (None)
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