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Theorem wunnat 17214
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 17157 . . 3 (𝜑 → (𝐶 Func 𝐷) ∈ 𝑈)
51, 4, 4wunxp 10134 . 2 (𝜑 → ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)) ∈ 𝑈)
6 df-hom 16577 . . . . . . 7 Hom = Slot 14
76, 1, 3wunstr 16495 . . . . . 6 (𝜑 → (Hom ‘𝐷) ∈ 𝑈)
81, 7wunrn 10139 . . . . 5 (𝜑 → ran (Hom ‘𝐷) ∈ 𝑈)
91, 8wununi 10116 . . . 4 (𝜑 ran (Hom ‘𝐷) ∈ 𝑈)
10 df-base 16477 . . . . 5 Base = Slot 1
1110, 1, 2wunstr 16495 . . . 4 (𝜑 → (Base‘𝐶) ∈ 𝑈)
121, 9, 11wunmap 10136 . . 3 (𝜑 → ( ran (Hom ‘𝐷) ↑m (Base‘𝐶)) ∈ 𝑈)
131, 12wunpw 10117 . 2 (𝜑 → 𝒫 ( ran (Hom ‘𝐷) ↑m (Base‘𝐶)) ∈ 𝑈)
14 fvex 6676 . . . . . 6 (1st𝑓) ∈ V
15 fvex 6676 . . . . . . . . 9 (1st𝑔) ∈ V
16 ovex 7178 . . . . . . . . . . . 12 ( ran (Hom ‘𝐷) ↑m (Base‘𝐶)) ∈ V
17 ssrab2 4053 . . . . . . . . . . . . 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 7181 . . . . . . . . . . . . . . . 16 ((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ⊆ ran (Hom ‘𝐷)
1918rgenw 3147 . . . . . . . . . . . . . . 15 𝑥 ∈ (Base‘𝐶)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ⊆ ran (Hom ‘𝐷)
20 ss2ixp 8462 . . . . . . . . . . . . . . 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 6676 . . . . . . . . . . . . . . 15 (Base‘𝐶) ∈ V
23 fvex 6676 . . . . . . . . . . . . . . . . 17 (Hom ‘𝐷) ∈ V
2423rnex 7606 . . . . . . . . . . . . . . . 16 ran (Hom ‘𝐷) ∈ V
2524uniex 7454 . . . . . . . . . . . . . . 15 ran (Hom ‘𝐷) ∈ V
2622, 25ixpconst 8459 . . . . . . . . . . . . . 14 X𝑥 ∈ (Base‘𝐶) ran (Hom ‘𝐷) = ( ran (Hom ‘𝐷) ↑m (Base‘𝐶))
2721, 26sseqtri 4000 . . . . . . . . . . . . 13 X𝑥 ∈ (Base‘𝐶)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ⊆ ( ran (Hom ‘𝐷) ↑m (Base‘𝐶))
2817, 27sstri 3973 . . . . . . . . . . . 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 5240 . . . . . . . . . . 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 3784 . . . . . . . . . 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 4362 . . . . . . . . . 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 3784 . . . . . . 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 4362 . . . . . . 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 3148 . . . 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 2818 . . . . . 6 (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
40 eqid 2818 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
41 eqid 2818 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
42 eqid 2818 . . . . . 6 (Hom ‘𝐷) = (Hom ‘𝐷)
43 eqid 2818 . . . . . 6 (comp‘𝐷) = (comp‘𝐷)
4439, 40, 41, 42, 43natfval 17204 . . . . 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 7755 . . . 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 10137 1 (𝜑 → (𝐶 Nat 𝐷) ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  wral 3135  {crab 3139  Vcvv 3492  [wsbc 3769  csb 3880  wss 3933  𝒫 cpw 4535  cop 4563   cuni 4830   × cxp 5546  ran crn 5549  wf 6344  cfv 6348  (class class class)co 7145  1st c1st 7676  2nd c2nd 7677  m cmap 8395  Xcixp 8449  WUnicwun 10110  1c1 10526  4c4 11682  cdc 12086  Basecbs 16471  Hom chom 16564  compcco 16565   Func cfunc 17112   Nat cnat 17199
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-map 8397  df-pm 8398  df-ixp 8450  df-wun 10112  df-slot 16475  df-base 16477  df-hom 16577  df-func 17116  df-nat 17201
This theorem is referenced by:  catcfuccl  17357
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