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Theorem catcfuccl 16806
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 2651 . . . . 5 (𝑋 Func 𝑌) = (𝑋 Func 𝑌)
3 eqid 2651 . . . . 5 (𝑋 Nat 𝑌) = (𝑋 Nat 𝑌)
4 eqid 2651 . . . . 5 (Base‘𝑋) = (Base‘𝑋)
5 eqid 2651 . . . . 5 (comp‘𝑌) = (comp‘𝑌)
6 inss2 3867 . . . . . 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 16794 . . . . . . 7 (𝜑𝐵 = (𝑈 ∩ Cat))
127, 11eleqtrd 2732 . . . . . 6 (𝜑𝑋 ∈ (𝑈 ∩ Cat))
136, 12sseldi 3634 . . . . 5 (𝜑𝑋 ∈ Cat)
14 catcfuccl.y . . . . . . 7 (𝜑𝑌𝐵)
1514, 11eleqtrd 2732 . . . . . 6 (𝜑𝑌 ∈ (𝑈 ∩ Cat))
166, 15sseldi 3634 . . . . 5 (𝜑𝑌 ∈ Cat)
17 eqidd 2652 . . . . 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 16665 . . . 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 15910 . . . . . . 7 Base = Slot 1
20 catcfuccl.1 . . . . . . . 8 (𝜑 → ω ∈ 𝑈)
2110, 20wunndx 15925 . . . . . . 7 (𝜑 → ndx ∈ 𝑈)
2219, 10, 21wunstr 15928 . . . . . 6 (𝜑 → (Base‘ndx) ∈ 𝑈)
23 inss1 3866 . . . . . . . 8 (𝑈 ∩ Cat) ⊆ 𝑈
2423, 12sseldi 3634 . . . . . . 7 (𝜑𝑋𝑈)
2523, 15sseldi 3634 . . . . . . 7 (𝜑𝑌𝑈)
2610, 24, 25wunfunc 16606 . . . . . 6 (𝜑 → (𝑋 Func 𝑌) ∈ 𝑈)
2710, 22, 26wunop 9582 . . . . 5 (𝜑 → ⟨(Base‘ndx), (𝑋 Func 𝑌)⟩ ∈ 𝑈)
28 df-hom 16013 . . . . . . 7 Hom = Slot 14
2928, 10, 21wunstr 15928 . . . . . 6 (𝜑 → (Hom ‘ndx) ∈ 𝑈)
3010, 24, 25wunnat 16663 . . . . . 6 (𝜑 → (𝑋 Nat 𝑌) ∈ 𝑈)
3110, 29, 30wunop 9582 . . . . 5 (𝜑 → ⟨(Hom ‘ndx), (𝑋 Nat 𝑌)⟩ ∈ 𝑈)
32 df-cco 16014 . . . . . . 7 comp = Slot 15
3332, 10, 21wunstr 15928 . . . . . 6 (𝜑 → (comp‘ndx) ∈ 𝑈)
3410, 26, 26wunxp 9584 . . . . . . . 8 (𝜑 → ((𝑋 Func 𝑌) × (𝑋 Func 𝑌)) ∈ 𝑈)
3510, 34, 26wunxp 9584 . . . . . . 7 (𝜑 → (((𝑋 Func 𝑌) × (𝑋 Func 𝑌)) × (𝑋 Func 𝑌)) ∈ 𝑈)
3632, 10, 25wunstr 15928 . . . . . . . . . . . . . 14 (𝜑 → (comp‘𝑌) ∈ 𝑈)
3710, 36wunrn 9589 . . . . . . . . . . . . 13 (𝜑 → ran (comp‘𝑌) ∈ 𝑈)
3810, 37wununi 9566 . . . . . . . . . . . 12 (𝜑 ran (comp‘𝑌) ∈ 𝑈)
3910, 38wunrn 9589 . . . . . . . . . . 11 (𝜑 → ran ran (comp‘𝑌) ∈ 𝑈)
4010, 39wununi 9566 . . . . . . . . . 10 (𝜑 ran ran (comp‘𝑌) ∈ 𝑈)
4110, 40wunpw 9567 . . . . . . . . 9 (𝜑 → 𝒫 ran ran (comp‘𝑌) ∈ 𝑈)
4219, 10, 24wunstr 15928 . . . . . . . . 9 (𝜑 → (Base‘𝑋) ∈ 𝑈)
4310, 41, 42wunmap 9586 . . . . . . . 8 (𝜑 → (𝒫 ran ran (comp‘𝑌) ↑𝑚 (Base‘𝑋)) ∈ 𝑈)
4410, 30wunrn 9589 . . . . . . . . . 10 (𝜑 → ran (𝑋 Nat 𝑌) ∈ 𝑈)
4510, 44wununi 9566 . . . . . . . . 9 (𝜑 ran (𝑋 Nat 𝑌) ∈ 𝑈)
4610, 45, 45wunxp 9584 . . . . . . . 8 (𝜑 → ( ran (𝑋 Nat 𝑌) × ran (𝑋 Nat 𝑌)) ∈ 𝑈)
4710, 43, 46wunpm 9585 . . . . . . 7 (𝜑 → ((𝒫 ran ran (comp‘𝑌) ↑𝑚 (Base‘𝑋)) ↑pm ( ran (𝑋 Nat 𝑌) × ran (𝑋 Nat 𝑌))) ∈ 𝑈)
48 fvex 6239 . . . . . . . . . . 11 (1st𝑣) ∈ V
49 fvex 6239 . . . . . . . . . . . . . 14 (2nd𝑣) ∈ V
50 ovex 6718 . . . . . . . . . . . . . . . . 17 (𝒫 ran ran (comp‘𝑌) ↑𝑚 (Base‘𝑋)) ∈ V
51 ovex 6718 . . . . . . . . . . . . . . . . . . . 20 (𝑋 Nat 𝑌) ∈ V
5251rnex 7142 . . . . . . . . . . . . . . . . . . 19 ran (𝑋 Nat 𝑌) ∈ V
5352uniex 6995 . . . . . . . . . . . . . . . . . 18 ran (𝑋 Nat 𝑌) ∈ V
5453, 53xpex 7004 . . . . . . . . . . . . . . . . 17 ( ran (𝑋 Nat 𝑌) × ran (𝑋 Nat 𝑌)) ∈ V
55 eqid 2651 . . . . . . . . . . . . . . . . . . . . 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 6721 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥)) ⊆ ran (⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))
57 ovssunirn 6721 . . . . . . . . . . . . . . . . . . . . . . . . 25 (⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥)) ⊆ ran (comp‘𝑌)
58 rnss 5386 . . . . . . . . . . . . . . . . . . . . . . . . 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 4490 . . . . . . . . . . . . . . . . . . . . . . . . 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 3645 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥)) ⊆ ran ran (comp‘𝑌)
62 ovex 6718 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥)) ∈ V
6362elpw 4197 . . . . . . . . . . . . . . . . . . . . . . 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 6423 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥))):(Base‘𝑋)⟶𝒫 ran ran (comp‘𝑌)
67 fvex 6239 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (comp‘𝑌) ∈ V
6867rnex 7142 . . . . . . . . . . . . . . . . . . . . . . . . 25 ran (comp‘𝑌) ∈ V
6968uniex 6995 . . . . . . . . . . . . . . . . . . . . . . . 24 ran (comp‘𝑌) ∈ V
7069rnex 7142 . . . . . . . . . . . . . . . . . . . . . . 23 ran ran (comp‘𝑌) ∈ V
7170uniex 6995 . . . . . . . . . . . . . . . . . . . . . 22 ran ran (comp‘𝑌) ∈ V
7271pwex 4878 . . . . . . . . . . . . . . . . . . . . 21 𝒫 ran ran (comp‘𝑌) ∈ V
73 fvex 6239 . . . . . . . . . . . . . . . . . . . . 21 (Base‘𝑋) ∈ V
7472, 73elmap 7928 . . . . . . . . . . . . . . . . . . . 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 2954 . . . . . . . . . . . . . . . . . 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 2651 . . . . . . . . . . . . . . . . . . 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 7282 . . . . . . . . . . . . . . . . . 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 6721 . . . . . . . . . . . . . . . . . 18 (𝑔(𝑋 Nat 𝑌)) ⊆ ran (𝑋 Nat 𝑌)
81 ovssunirn 6721 . . . . . . . . . . . . . . . . . 18 (𝑓(𝑋 Nat 𝑌)𝑔) ⊆ ran (𝑋 Nat 𝑌)
82 xpss12 5158 . . . . . . . . . . . . . . . . . 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 708 . . . . . . . . . . . . . . . . 17 ((𝑔(𝑋 Nat 𝑌)) × (𝑓(𝑋 Nat 𝑌)𝑔)) ⊆ ( ran (𝑋 Nat 𝑌) × ran (𝑋 Nat 𝑌))
84 elpm2r 7917 . . . . . . . . . . . . . . . . 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 709 . . . . . . . . . . . . . . . 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 3483 . . . . . . . . . . . . . . 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 4020 . . . . . . . . . . . . . . 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 3483 . . . . . . . . . . . 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 4020 . . . . . . . . . . . 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 2954 . . . . . . . . 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 2651 . . . . . . . . . 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 7282 . . . . . . . . 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 9587 . . . . . 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 9582 . . . . 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 9571 . . . 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 2730 . . 3 (𝜑𝑄𝑈)
1031, 13, 16fuccat 16677 . . 3 (𝜑𝑄 ∈ Cat)
104102, 103elind 3831 . 2 (𝜑𝑄 ∈ (𝑈 ∩ Cat))
105104, 11eleqtrrd 2733 1 (𝜑𝑄𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  [wsbc 3468  csb 3566  cin 3606  wss 3607  𝒫 cpw 4191  {ctp 4214  cop 4216   cuni 4468  cmpt 4762   × cxp 5141  ran crn 5144  wf 5922  cfv 5926  (class class class)co 6690  cmpt2 6692  ωcom 7107  1st c1st 7208  2nd c2nd 7209  𝑚 cmap 7899  pm cpm 7900  WUnicwun 9560  1c1 9975  4c4 11110  5c5 11111  cdc 11531  ndxcnx 15901  Basecbs 15904  Hom chom 15999  compcco 16000  Catccat 16372   Func cfunc 16561   Nat cnat 16648   FuncCat cfuc 16649  CatCatccatc 16791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-omul 7610  df-er 7787  df-ec 7789  df-qs 7793  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-wun 9562  df-ni 9732  df-pli 9733  df-mi 9734  df-lti 9735  df-plpq 9768  df-mpq 9769  df-ltpq 9770  df-enq 9771  df-nq 9772  df-erq 9773  df-plq 9774  df-mq 9775  df-1nq 9776  df-rq 9777  df-ltnq 9778  df-np 9841  df-plp 9843  df-ltp 9845  df-enr 9915  df-nr 9916  df-c 9980  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-fz 12365  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-hom 16013  df-cco 16014  df-cat 16376  df-cid 16377  df-func 16565  df-nat 16650  df-fuc 16651  df-catc 16792
This theorem is referenced by: (None)
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