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Theorem wunnat 17218
Description: A weak universe is closed under the natural transformation operation. (Contributed by Mario Carneiro, 12-Jan-2017.)
Hypotheses
Ref Expression
wunnat.1 (𝜑𝑈 ∈ WUni)
wunnat.2 (𝜑𝐶𝑈)
wunnat.3 (𝜑𝐷𝑈)
Assertion
Ref Expression
wunnat (𝜑 → (𝐶 Nat 𝐷) ∈ 𝑈)

Proof of Theorem wunnat
Dummy variables 𝑓 𝑎 𝑔 𝑟 𝑠 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wunnat.1 . 2 (𝜑𝑈 ∈ WUni)
2 wunnat.2 . . . 4 (𝜑𝐶𝑈)
3 wunnat.3 . . . 4 (𝜑𝐷𝑈)
41, 2, 3wunfunc 17161 . . 3 (𝜑 → (𝐶 Func 𝐷) ∈ 𝑈)
51, 4, 4wunxp 10138 . 2 (𝜑 → ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)) ∈ 𝑈)
6 df-hom 16581 . . . . . . 7 Hom = Slot 14
76, 1, 3wunstr 16499 . . . . . 6 (𝜑 → (Hom ‘𝐷) ∈ 𝑈)
81, 7wunrn 10143 . . . . 5 (𝜑 → ran (Hom ‘𝐷) ∈ 𝑈)
91, 8wununi 10120 . . . 4 (𝜑 ran (Hom ‘𝐷) ∈ 𝑈)
10 df-base 16481 . . . . 5 Base = Slot 1
1110, 1, 2wunstr 16499 . . . 4 (𝜑 → (Base‘𝐶) ∈ 𝑈)
121, 9, 11wunmap 10140 . . 3 (𝜑 → ( ran (Hom ‘𝐷) ↑m (Base‘𝐶)) ∈ 𝑈)
131, 12wunpw 10121 . 2 (𝜑 → 𝒫 ( ran (Hom ‘𝐷) ↑m (Base‘𝐶)) ∈ 𝑈)
14 fvex 6679 . . . . . 6 (1st𝑓) ∈ V
15 fvex 6679 . . . . . . . . 9 (1st𝑔) ∈ V
16 ovex 7184 . . . . . . . . . . . 12 ( ran (Hom ‘𝐷) ↑m (Base‘𝐶)) ∈ V
17 ssrab2 4059 . . . . . . . . . . . . 13 {𝑎X𝑥 ∈ (Base‘𝐶)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐷)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘𝑧)) = (((𝑥(2nd𝑔)𝑦)‘𝑧)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐷)(𝑠𝑦))(𝑎𝑥))} ⊆ X𝑥 ∈ (Base‘𝐶)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥))
18 ovssunirn 7187 . . . . . . . . . . . . . . . 16 ((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ⊆ ran (Hom ‘𝐷)
1918rgenw 3154 . . . . . . . . . . . . . . 15 𝑥 ∈ (Base‘𝐶)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ⊆ ran (Hom ‘𝐷)
20 ss2ixp 8466 . . . . . . . . . . . . . . 15 (∀𝑥 ∈ (Base‘𝐶)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ⊆ ran (Hom ‘𝐷) → X𝑥 ∈ (Base‘𝐶)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ⊆ X𝑥 ∈ (Base‘𝐶) ran (Hom ‘𝐷))
2119, 20ax-mp 5 . . . . . . . . . . . . . 14 X𝑥 ∈ (Base‘𝐶)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ⊆ X𝑥 ∈ (Base‘𝐶) ran (Hom ‘𝐷)
22 fvex 6679 . . . . . . . . . . . . . . 15 (Base‘𝐶) ∈ V
23 fvex 6679 . . . . . . . . . . . . . . . . 17 (Hom ‘𝐷) ∈ V
2423rnex 7608 . . . . . . . . . . . . . . . 16 ran (Hom ‘𝐷) ∈ V
2524uniex 7458 . . . . . . . . . . . . . . 15 ran (Hom ‘𝐷) ∈ V
2622, 25ixpconst 8463 . . . . . . . . . . . . . 14 X𝑥 ∈ (Base‘𝐶) ran (Hom ‘𝐷) = ( ran (Hom ‘𝐷) ↑m (Base‘𝐶))
2721, 26sseqtri 4006 . . . . . . . . . . . . 13 X𝑥 ∈ (Base‘𝐶)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ⊆ ( ran (Hom ‘𝐷) ↑m (Base‘𝐶))
2817, 27sstri 3979 . . . . . . . . . . . 12 {𝑎X𝑥 ∈ (Base‘𝐶)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐷)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘𝑧)) = (((𝑥(2nd𝑔)𝑦)‘𝑧)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐷)(𝑠𝑦))(𝑎𝑥))} ⊆ ( ran (Hom ‘𝐷) ↑m (Base‘𝐶))
2916, 28elpwi2 5245 . . . . . . . . . . 11 {𝑎X𝑥 ∈ (Base‘𝐶)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐷)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘𝑧)) = (((𝑥(2nd𝑔)𝑦)‘𝑧)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐷)(𝑠𝑦))(𝑎𝑥))} ∈ 𝒫 ( ran (Hom ‘𝐷) ↑m (Base‘𝐶))
3029sbcth 3790 . . . . . . . . . 10 ((1st𝑔) ∈ V → [(1st𝑔) / 𝑠]{𝑎X𝑥 ∈ (Base‘𝐶)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐷)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘𝑧)) = (((𝑥(2nd𝑔)𝑦)‘𝑧)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐷)(𝑠𝑦))(𝑎𝑥))} ∈ 𝒫 ( ran (Hom ‘𝐷) ↑m (Base‘𝐶)))
31 sbcel1g 4368 . . . . . . . . . 10 ((1st𝑔) ∈ V → ([(1st𝑔) / 𝑠]{𝑎X𝑥 ∈ (Base‘𝐶)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐷)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘𝑧)) = (((𝑥(2nd𝑔)𝑦)‘𝑧)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐷)(𝑠𝑦))(𝑎𝑥))} ∈ 𝒫 ( ran (Hom ‘𝐷) ↑m (Base‘𝐶)) ↔ (1st𝑔) / 𝑠{𝑎X𝑥 ∈ (Base‘𝐶)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐷)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘𝑧)) = (((𝑥(2nd𝑔)𝑦)‘𝑧)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐷)(𝑠𝑦))(𝑎𝑥))} ∈ 𝒫 ( ran (Hom ‘𝐷) ↑m (Base‘𝐶))))
3230, 31mpbid 233 . . . . . . . . 9 ((1st𝑔) ∈ V → (1st𝑔) / 𝑠{𝑎X𝑥 ∈ (Base‘𝐶)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐷)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘𝑧)) = (((𝑥(2nd𝑔)𝑦)‘𝑧)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐷)(𝑠𝑦))(𝑎𝑥))} ∈ 𝒫 ( ran (Hom ‘𝐷) ↑m (Base‘𝐶)))
3315, 32ax-mp 5 . . . . . . . 8 (1st𝑔) / 𝑠{𝑎X𝑥 ∈ (Base‘𝐶)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐷)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘𝑧)) = (((𝑥(2nd𝑔)𝑦)‘𝑧)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐷)(𝑠𝑦))(𝑎𝑥))} ∈ 𝒫 ( ran (Hom ‘𝐷) ↑m (Base‘𝐶))
3433sbcth 3790 . . . . . . 7 ((1st𝑓) ∈ V → [(1st𝑓) / 𝑟](1st𝑔) / 𝑠{𝑎X𝑥 ∈ (Base‘𝐶)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐷)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘𝑧)) = (((𝑥(2nd𝑔)𝑦)‘𝑧)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐷)(𝑠𝑦))(𝑎𝑥))} ∈ 𝒫 ( ran (Hom ‘𝐷) ↑m (Base‘𝐶)))
35 sbcel1g 4368 . . . . . . 7 ((1st𝑓) ∈ V → ([(1st𝑓) / 𝑟](1st𝑔) / 𝑠{𝑎X𝑥 ∈ (Base‘𝐶)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐷)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘𝑧)) = (((𝑥(2nd𝑔)𝑦)‘𝑧)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐷)(𝑠𝑦))(𝑎𝑥))} ∈ 𝒫 ( ran (Hom ‘𝐷) ↑m (Base‘𝐶)) ↔ (1st𝑓) / 𝑟(1st𝑔) / 𝑠{𝑎X𝑥 ∈ (Base‘𝐶)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐷)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘𝑧)) = (((𝑥(2nd𝑔)𝑦)‘𝑧)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐷)(𝑠𝑦))(𝑎𝑥))} ∈ 𝒫 ( ran (Hom ‘𝐷) ↑m (Base‘𝐶))))
3634, 35mpbid 233 . . . . . 6 ((1st𝑓) ∈ V → (1st𝑓) / 𝑟(1st𝑔) / 𝑠{𝑎X𝑥 ∈ (Base‘𝐶)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐷)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘𝑧)) = (((𝑥(2nd𝑔)𝑦)‘𝑧)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐷)(𝑠𝑦))(𝑎𝑥))} ∈ 𝒫 ( ran (Hom ‘𝐷) ↑m (Base‘𝐶)))
3714, 36ax-mp 5 . . . . 5 (1st𝑓) / 𝑟(1st𝑔) / 𝑠{𝑎X𝑥 ∈ (Base‘𝐶)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐷)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘𝑧)) = (((𝑥(2nd𝑔)𝑦)‘𝑧)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐷)(𝑠𝑦))(𝑎𝑥))} ∈ 𝒫 ( ran (Hom ‘𝐷) ↑m (Base‘𝐶))
3837rgen2w 3155 . . . 4 𝑓 ∈ (𝐶 Func 𝐷)∀𝑔 ∈ (𝐶 Func 𝐷)(1st𝑓) / 𝑟(1st𝑔) / 𝑠{𝑎X𝑥 ∈ (Base‘𝐶)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐷)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘𝑧)) = (((𝑥(2nd𝑔)𝑦)‘𝑧)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐷)(𝑠𝑦))(𝑎𝑥))} ∈ 𝒫 ( ran (Hom ‘𝐷) ↑m (Base‘𝐶))
39 eqid 2825 . . . . . 6 (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
40 eqid 2825 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
41 eqid 2825 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
42 eqid 2825 . . . . . 6 (Hom ‘𝐷) = (Hom ‘𝐷)
43 eqid 2825 . . . . . 6 (comp‘𝐷) = (comp‘𝐷)
4439, 40, 41, 42, 43natfval 17208 . . . . 5 (𝐶 Nat 𝐷) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑔 ∈ (𝐶 Func 𝐷) ↦ (1st𝑓) / 𝑟(1st𝑔) / 𝑠{𝑎X𝑥 ∈ (Base‘𝐶)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐷)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘𝑧)) = (((𝑥(2nd𝑔)𝑦)‘𝑧)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐷)(𝑠𝑦))(𝑎𝑥))})
4544fmpo 7760 . . . 4 (∀𝑓 ∈ (𝐶 Func 𝐷)∀𝑔 ∈ (𝐶 Func 𝐷)(1st𝑓) / 𝑟(1st𝑔) / 𝑠{𝑎X𝑥 ∈ (Base‘𝐶)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐷)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘𝑧)) = (((𝑥(2nd𝑔)𝑦)‘𝑧)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐷)(𝑠𝑦))(𝑎𝑥))} ∈ 𝒫 ( ran (Hom ‘𝐷) ↑m (Base‘𝐶)) ↔ (𝐶 Nat 𝐷):((𝐶 Func 𝐷) × (𝐶 Func 𝐷))⟶𝒫 ( ran (Hom ‘𝐷) ↑m (Base‘𝐶)))
4638, 45mpbi 231 . . 3 (𝐶 Nat 𝐷):((𝐶 Func 𝐷) × (𝐶 Func 𝐷))⟶𝒫 ( ran (Hom ‘𝐷) ↑m (Base‘𝐶))
4746a1i 11 . 2 (𝜑 → (𝐶 Nat 𝐷):((𝐶 Func 𝐷) × (𝐶 Func 𝐷))⟶𝒫 ( ran (Hom ‘𝐷) ↑m (Base‘𝐶)))
481, 5, 13, 47wunf 10141 1 (𝜑 → (𝐶 Nat 𝐷) ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530  wcel 2107  wral 3142  {crab 3146  Vcvv 3499  [wsbc 3775  csb 3886  wss 3939  𝒫 cpw 4541  cop 4569   cuni 4836   × cxp 5551  ran crn 5554  wf 6347  cfv 6351  (class class class)co 7151  1st c1st 7681  2nd c2nd 7682  m cmap 8399  Xcixp 8453  WUnicwun 10114  1c1 10530  4c4 11686  cdc 12090  Basecbs 16475  Hom chom 16568  compcco 16569   Func cfunc 17116   Nat cnat 17203
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 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-1st 7683  df-2nd 7684  df-map 8401  df-pm 8402  df-ixp 8454  df-wun 10116  df-slot 16479  df-base 16481  df-hom 16581  df-func 17120  df-nat 17205
This theorem is referenced by:  catcfuccl  17361
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