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Theorem catcfuccl 17359
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 2826 . . . . 5 (𝑋 Func 𝑌) = (𝑋 Func 𝑌)
3 eqid 2826 . . . . 5 (𝑋 Nat 𝑌) = (𝑋 Nat 𝑌)
4 eqid 2826 . . . . 5 (Base‘𝑋) = (Base‘𝑋)
5 eqid 2826 . . . . 5 (comp‘𝑌) = (comp‘𝑌)
6 catcfuccl.x . . . . . . 7 (𝜑𝑋𝐵)
7 catcfuccl.c . . . . . . . 8 𝐶 = (CatCat‘𝑈)
8 catcfuccl.b . . . . . . . 8 𝐵 = (Base‘𝐶)
9 catcfuccl.u . . . . . . . 8 (𝜑𝑈 ∈ WUni)
107, 8, 9catcbas 17347 . . . . . . 7 (𝜑𝐵 = (𝑈 ∩ Cat))
116, 10eleqtrd 2920 . . . . . 6 (𝜑𝑋 ∈ (𝑈 ∩ Cat))
1211elin2d 4180 . . . . 5 (𝜑𝑋 ∈ Cat)
13 catcfuccl.y . . . . . . 7 (𝜑𝑌𝐵)
1413, 10eleqtrd 2920 . . . . . 6 (𝜑𝑌 ∈ (𝑈 ∩ Cat))
1514elin2d 4180 . . . . 5 (𝜑𝑌 ∈ Cat)
16 eqidd 2827 . . . . 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))))))
171, 2, 3, 4, 5, 12, 15, 16fucval 17218 . . . 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)))))⟩})
18 df-base 16479 . . . . . . 7 Base = Slot 1
19 catcfuccl.1 . . . . . . . 8 (𝜑 → ω ∈ 𝑈)
209, 19wunndx 16494 . . . . . . 7 (𝜑 → ndx ∈ 𝑈)
2118, 9, 20wunstr 16497 . . . . . 6 (𝜑 → (Base‘ndx) ∈ 𝑈)
2211elin1d 4179 . . . . . . 7 (𝜑𝑋𝑈)
2314elin1d 4179 . . . . . . 7 (𝜑𝑌𝑈)
249, 22, 23wunfunc 17159 . . . . . 6 (𝜑 → (𝑋 Func 𝑌) ∈ 𝑈)
259, 21, 24wunop 10133 . . . . 5 (𝜑 → ⟨(Base‘ndx), (𝑋 Func 𝑌)⟩ ∈ 𝑈)
26 df-hom 16579 . . . . . . 7 Hom = Slot 14
2726, 9, 20wunstr 16497 . . . . . 6 (𝜑 → (Hom ‘ndx) ∈ 𝑈)
289, 22, 23wunnat 17216 . . . . . 6 (𝜑 → (𝑋 Nat 𝑌) ∈ 𝑈)
299, 27, 28wunop 10133 . . . . 5 (𝜑 → ⟨(Hom ‘ndx), (𝑋 Nat 𝑌)⟩ ∈ 𝑈)
30 df-cco 16580 . . . . . . 7 comp = Slot 15
3130, 9, 20wunstr 16497 . . . . . 6 (𝜑 → (comp‘ndx) ∈ 𝑈)
329, 24, 24wunxp 10135 . . . . . . . 8 (𝜑 → ((𝑋 Func 𝑌) × (𝑋 Func 𝑌)) ∈ 𝑈)
339, 32, 24wunxp 10135 . . . . . . 7 (𝜑 → (((𝑋 Func 𝑌) × (𝑋 Func 𝑌)) × (𝑋 Func 𝑌)) ∈ 𝑈)
3430, 9, 23wunstr 16497 . . . . . . . . . . . . . 14 (𝜑 → (comp‘𝑌) ∈ 𝑈)
359, 34wunrn 10140 . . . . . . . . . . . . 13 (𝜑 → ran (comp‘𝑌) ∈ 𝑈)
369, 35wununi 10117 . . . . . . . . . . . 12 (𝜑 ran (comp‘𝑌) ∈ 𝑈)
379, 36wunrn 10140 . . . . . . . . . . 11 (𝜑 → ran ran (comp‘𝑌) ∈ 𝑈)
389, 37wununi 10117 . . . . . . . . . 10 (𝜑 ran ran (comp‘𝑌) ∈ 𝑈)
399, 38wunpw 10118 . . . . . . . . 9 (𝜑 → 𝒫 ran ran (comp‘𝑌) ∈ 𝑈)
4018, 9, 22wunstr 16497 . . . . . . . . 9 (𝜑 → (Base‘𝑋) ∈ 𝑈)
419, 39, 40wunmap 10137 . . . . . . . 8 (𝜑 → (𝒫 ran ran (comp‘𝑌) ↑m (Base‘𝑋)) ∈ 𝑈)
429, 28wunrn 10140 . . . . . . . . . 10 (𝜑 → ran (𝑋 Nat 𝑌) ∈ 𝑈)
439, 42wununi 10117 . . . . . . . . 9 (𝜑 ran (𝑋 Nat 𝑌) ∈ 𝑈)
449, 43, 43wunxp 10135 . . . . . . . 8 (𝜑 → ( ran (𝑋 Nat 𝑌) × ran (𝑋 Nat 𝑌)) ∈ 𝑈)
459, 41, 44wunpm 10136 . . . . . . 7 (𝜑 → ((𝒫 ran ran (comp‘𝑌) ↑m (Base‘𝑋)) ↑pm ( ran (𝑋 Nat 𝑌) × ran (𝑋 Nat 𝑌))) ∈ 𝑈)
46 fvex 6680 . . . . . . . . . . 11 (1st𝑣) ∈ V
47 fvex 6680 . . . . . . . . . . . . . 14 (2nd𝑣) ∈ V
48 ovex 7181 . . . . . . . . . . . . . . . . 17 (𝒫 ran ran (comp‘𝑌) ↑m (Base‘𝑋)) ∈ V
49 ovex 7181 . . . . . . . . . . . . . . . . . . . 20 (𝑋 Nat 𝑌) ∈ V
5049rnex 7605 . . . . . . . . . . . . . . . . . . 19 ran (𝑋 Nat 𝑌) ∈ V
5150uniex 7455 . . . . . . . . . . . . . . . . . 18 ran (𝑋 Nat 𝑌) ∈ V
5251, 51xpex 7465 . . . . . . . . . . . . . . . . 17 ( ran (𝑋 Nat 𝑌) × ran (𝑋 Nat 𝑌)) ∈ V
53 eqid 2826 . . . . . . . . . . . . . . . . . . . . 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)))
54 ovssunirn 7184 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥)) ⊆ ran (⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))
55 ovssunirn 7184 . . . . . . . . . . . . . . . . . . . . . . . . 25 (⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥)) ⊆ ran (comp‘𝑌)
56 rnss 5808 . . . . . . . . . . . . . . . . . . . . . . . . 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))
57 uniss 4858 . . . . . . . . . . . . . . . . . . . . . . . . 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))
5855, 56, 57mp2b 10 . . . . . . . . . . . . . . . . . . . . . . . 24 ran (⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥)) ⊆ ran ran (comp‘𝑌)
5954, 58sstri 3980 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥)) ⊆ ran ran (comp‘𝑌)
60 ovex 7181 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥)) ∈ V
6160elpw 4549 . . . . . . . . . . . . . . . . . . . . . . 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))
6259, 61mpbir 232 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥)) ∈ 𝒫 ran ran (comp‘𝑌)
6362a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (Base‘𝑋) → ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥)) ∈ 𝒫 ran ran (comp‘𝑌))
6453, 63fmpti 6872 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥))):(Base‘𝑋)⟶𝒫 ran ran (comp‘𝑌)
65 fvex 6680 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (comp‘𝑌) ∈ V
6665rnex 7605 . . . . . . . . . . . . . . . . . . . . . . . . 25 ran (comp‘𝑌) ∈ V
6766uniex 7455 . . . . . . . . . . . . . . . . . . . . . . . 24 ran (comp‘𝑌) ∈ V
6867rnex 7605 . . . . . . . . . . . . . . . . . . . . . . 23 ran ran (comp‘𝑌) ∈ V
6968uniex 7455 . . . . . . . . . . . . . . . . . . . . . 22 ran ran (comp‘𝑌) ∈ V
7069pwex 5278 . . . . . . . . . . . . . . . . . . . . 21 𝒫 ran ran (comp‘𝑌) ∈ V
71 fvex 6680 . . . . . . . . . . . . . . . . . . . . 21 (Base‘𝑋) ∈ V
7270, 71elmap 8425 . . . . . . . . . . . . . . . . . . . 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))
7364, 72mpbir 232 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑌)((1st)‘𝑥))(𝑎𝑥))) ∈ (𝒫 ran ran (comp‘𝑌) ↑m (Base‘𝑋))
7473rgen2w 3156 . . . . . . . . . . . . . . . . . 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))
75 eqid 2826 . . . . . . . . . . . . . . . . . . 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))))
7675fmpo 7757 . . . . . . . . . . . . . . . . . 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)))
7774, 76mpbi 231 . . . . . . . . . . . . . . . . 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))
78 ovssunirn 7184 . . . . . . . . . . . . . . . . . 18 (𝑔(𝑋 Nat 𝑌)) ⊆ ran (𝑋 Nat 𝑌)
79 ovssunirn 7184 . . . . . . . . . . . . . . . . . 18 (𝑓(𝑋 Nat 𝑌)𝑔) ⊆ ran (𝑋 Nat 𝑌)
80 xpss12 5569 . . . . . . . . . . . . . . . . . 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)))
8178, 79, 80mp2an 688 . . . . . . . . . . . . . . . . 17 ((𝑔(𝑋 Nat 𝑌)) × (𝑓(𝑋 Nat 𝑌)𝑔)) ⊆ ( ran (𝑋 Nat 𝑌) × ran (𝑋 Nat 𝑌))
82 elpm2r 8414 . . . . . . . . . . . . . . . . 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))))
8348, 52, 77, 81, 82mp4an 689 . . . . . . . . . . . . . . . 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)))
8483sbcth 3791 . . . . . . . . . . . . . . 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))))
85 sbcel1g 4369 . . . . . . . . . . . . . . 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)))))
8684, 85mpbid 233 . . . . . . . . . . . . . 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))))
8747, 86ax-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)))
8887sbcth 3791 . . . . . . . . . . . 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))))
89 sbcel1g 4369 . . . . . . . . . . . 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)))))
9088, 89mpbid 233 . . . . . . . . . . 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))))
9146, 90ax-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)))
9291rgen2w 3156 . . . . . . . . 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)))
93 eqid 2826 . . . . . . . . . 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)))))
9493fmpo 7757 . . . . . . . . 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))))
9592, 94mpbi 231 . . . . . . . 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)))
9695a1i 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))))
979, 33, 45, 96wunf 10138 . . . . . 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)
989, 31, 97wunop 10133 . . . . 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)
999, 25, 29, 98wuntp 10122 . . . 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)
10017, 99eqeltrd 2918 . . 3 (𝜑𝑄𝑈)
1011, 12, 15fuccat 17230 . . 3 (𝜑𝑄 ∈ Cat)
102100, 101elind 4175 . 2 (𝜑𝑄 ∈ (𝑈 ∩ Cat))
103102, 10eleqtrrd 2921 1 (𝜑𝑄𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530  wcel 2107  wral 3143  Vcvv 3500  [wsbc 3776  csb 3887  cin 3939  wss 3940  𝒫 cpw 4542  {ctp 4568  cop 4570   cuni 4837  cmpt 5143   × cxp 5552  ran crn 5555  wf 6348  cfv 6352  (class class class)co 7148  cmpo 7150  ωcom 7568  1st c1st 7678  2nd c2nd 7679  m cmap 8396  pm cpm 8397  WUnicwun 10111  1c1 10527  4c4 11683  5c5 11684  cdc 12087  ndxcnx 16470  Basecbs 16473  Hom chom 16566  compcco 16567  Catccat 16925   Func cfunc 17114   Nat cnat 17201   FuncCat cfuc 17202  CatCatccatc 17344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-inf2 9093  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 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-oadd 8097  df-omul 8098  df-er 8279  df-ec 8281  df-qs 8285  df-map 8398  df-pm 8399  df-ixp 8451  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-wun 10113  df-ni 10283  df-pli 10284  df-mi 10285  df-lti 10286  df-plpq 10319  df-mpq 10320  df-ltpq 10321  df-enq 10322  df-nq 10323  df-erq 10324  df-plq 10325  df-mq 10326  df-1nq 10327  df-rq 10328  df-ltnq 10329  df-np 10392  df-plp 10394  df-ltp 10396  df-enr 10466  df-nr 10467  df-c 10532  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11628  df-2 11689  df-3 11690  df-4 11691  df-5 11692  df-6 11693  df-7 11694  df-8 11695  df-9 11696  df-n0 11887  df-z 11971  df-dec 12088  df-uz 12233  df-fz 12883  df-struct 16475  df-ndx 16476  df-slot 16477  df-base 16479  df-hom 16579  df-cco 16580  df-cat 16929  df-cid 16930  df-func 17118  df-nat 17203  df-fuc 17204  df-catc 17345
This theorem is referenced by: (None)
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