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Theorem wunnat 18010
Description: A weak universe is closed under the natural transformation operation. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof shortened by AV, 13-Oct-2024.)
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 17951 . . 3 (𝜑 → (𝐶 Func 𝐷) ∈ 𝑈)
51, 4, 4wunxp 10761 . 2 (𝜑 → ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)) ∈ 𝑈)
6 homid 17457 . . . . . . 7 Hom = Slot (Hom ‘ndx)
76, 1, 3wunstr 17221 . . . . . 6 (𝜑 → (Hom ‘𝐷) ∈ 𝑈)
81, 7wunrn 10766 . . . . 5 (𝜑 → ran (Hom ‘𝐷) ∈ 𝑈)
91, 8wununi 10743 . . . 4 (𝜑 ran (Hom ‘𝐷) ∈ 𝑈)
10 baseid 17247 . . . . 5 Base = Slot (Base‘ndx)
1110, 1, 2wunstr 17221 . . . 4 (𝜑 → (Base‘𝐶) ∈ 𝑈)
121, 9, 11wunmap 10763 . . 3 (𝜑 → ( ran (Hom ‘𝐷) ↑m (Base‘𝐶)) ∈ 𝑈)
131, 12wunpw 10744 . 2 (𝜑 → 𝒫 ( ran (Hom ‘𝐷) ↑m (Base‘𝐶)) ∈ 𝑈)
14 fvex 6919 . . . . . 6 (1st𝑓) ∈ V
15 fvex 6919 . . . . . . . . 9 (1st𝑔) ∈ V
16 ovex 7463 . . . . . . . . . . . 12 ( ran (Hom ‘𝐷) ↑m (Base‘𝐶)) ∈ V
17 ssrab2 4089 . . . . . . . . . . . . 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 7466 . . . . . . . . . . . . . . . 16 ((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ⊆ ran (Hom ‘𝐷)
1918rgenw 3062 . . . . . . . . . . . . . . 15 𝑥 ∈ (Base‘𝐶)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ⊆ ran (Hom ‘𝐷)
20 ss2ixp 8948 . . . . . . . . . . . . . . 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 6919 . . . . . . . . . . . . . . 15 (Base‘𝐶) ∈ V
23 fvex 6919 . . . . . . . . . . . . . . . . 17 (Hom ‘𝐷) ∈ V
2423rnex 7932 . . . . . . . . . . . . . . . 16 ran (Hom ‘𝐷) ∈ V
2524uniex 7759 . . . . . . . . . . . . . . 15 ran (Hom ‘𝐷) ∈ V
2622, 25ixpconst 8945 . . . . . . . . . . . . . 14 X𝑥 ∈ (Base‘𝐶) ran (Hom ‘𝐷) = ( ran (Hom ‘𝐷) ↑m (Base‘𝐶))
2721, 26sseqtri 4031 . . . . . . . . . . . . 13 X𝑥 ∈ (Base‘𝐶)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ⊆ ( ran (Hom ‘𝐷) ↑m (Base‘𝐶))
2817, 27sstri 4004 . . . . . . . . . . . 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 5340 . . . . . . . . . . 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 3805 . . . . . . . . . 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 4421 . . . . . . . . . 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 232 . . . . . . . . 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 3805 . . . . . . 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 4421 . . . . . . 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 232 . . . . . 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 3063 . . . 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 2734 . . . . . 6 (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
40 eqid 2734 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
41 eqid 2734 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
42 eqid 2734 . . . . . 6 (Hom ‘𝐷) = (Hom ‘𝐷)
43 eqid 2734 . . . . . 6 (comp‘𝐷) = (comp‘𝐷)
4439, 40, 41, 42, 43natfval 18000 . . . . 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 8091 . . . 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 230 . . 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 10764 1 (𝜑 → (𝐶 Nat 𝐷) ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1536  wcel 2105  wral 3058  {crab 3432  Vcvv 3477  [wsbc 3790  csb 3907  wss 3962  𝒫 cpw 4604  cop 4636   cuni 4911   × cxp 5686  ran crn 5689  wf 6558  cfv 6562  (class class class)co 7430  1st c1st 8010  2nd c2nd 8011  m cmap 8864  Xcixp 8935  WUnicwun 10737  ndxcnx 17226  Basecbs 17244  Hom chom 17308  compcco 17309   Func cfunc 17904   Nat cnat 17995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-er 8743  df-map 8866  df-pm 8867  df-ixp 8936  df-en 8984  df-dom 8985  df-sdom 8986  df-wun 10739  df-pnf 11294  df-mnf 11295  df-ltxr 11297  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-dec 12731  df-slot 17215  df-ndx 17227  df-base 17245  df-hom 17321  df-func 17908  df-nat 17997
This theorem is referenced by:  catcfuccl  18172  catcfucclOLD  18173
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