MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wunnat Structured version   Visualization version   GIF version

Theorem wunnat 18004
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 17946 . . 3 (𝜑 → (𝐶 Func 𝐷) ∈ 𝑈)
51, 4, 4wunxp 10764 . 2 (𝜑 → ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)) ∈ 𝑈)
6 homid 17456 . . . . . . 7 Hom = Slot (Hom ‘ndx)
76, 1, 3wunstr 17225 . . . . . 6 (𝜑 → (Hom ‘𝐷) ∈ 𝑈)
81, 7wunrn 10769 . . . . 5 (𝜑 → ran (Hom ‘𝐷) ∈ 𝑈)
91, 8wununi 10746 . . . 4 (𝜑 ran (Hom ‘𝐷) ∈ 𝑈)
10 baseid 17250 . . . . 5 Base = Slot (Base‘ndx)
1110, 1, 2wunstr 17225 . . . 4 (𝜑 → (Base‘𝐶) ∈ 𝑈)
121, 9, 11wunmap 10766 . . 3 (𝜑 → ( ran (Hom ‘𝐷) ↑m (Base‘𝐶)) ∈ 𝑈)
131, 12wunpw 10747 . 2 (𝜑 → 𝒫 ( ran (Hom ‘𝐷) ↑m (Base‘𝐶)) ∈ 𝑈)
14 fvex 6919 . . . . . 6 (1st𝑓) ∈ V
15 fvex 6919 . . . . . . . . 9 (1st𝑔) ∈ V
16 ovex 7464 . . . . . . . . . . . 12 ( ran (Hom ‘𝐷) ↑m (Base‘𝐶)) ∈ V
17 ssrab2 4080 . . . . . . . . . . . . 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 7467 . . . . . . . . . . . . . . . 16 ((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ⊆ ran (Hom ‘𝐷)
1918rgenw 3065 . . . . . . . . . . . . . . 15 𝑥 ∈ (Base‘𝐶)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ⊆ ran (Hom ‘𝐷)
20 ss2ixp 8950 . . . . . . . . . . . . . . 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 7761 . . . . . . . . . . . . . . 15 ran (Hom ‘𝐷) ∈ V
2622, 25ixpconst 8947 . . . . . . . . . . . . . 14 X𝑥 ∈ (Base‘𝐶) ran (Hom ‘𝐷) = ( ran (Hom ‘𝐷) ↑m (Base‘𝐶))
2721, 26sseqtri 4032 . . . . . . . . . . . . 13 X𝑥 ∈ (Base‘𝐶)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ⊆ ( ran (Hom ‘𝐷) ↑m (Base‘𝐶))
2817, 27sstri 3993 . . . . . . . . . . . 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 5335 . . . . . . . . . . 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 3803 . . . . . . . . . 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 4416 . . . . . . . . . 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 3803 . . . . . . 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 4416 . . . . . . 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 3066 . . . 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 2737 . . . . . 6 (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
40 eqid 2737 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
41 eqid 2737 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
42 eqid 2737 . . . . . 6 (Hom ‘𝐷) = (Hom ‘𝐷)
43 eqid 2737 . . . . . 6 (comp‘𝐷) = (comp‘𝐷)
4439, 40, 41, 42, 43natfval 17994 . . . . 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 8093 . . . 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 10767 1 (𝜑 → (𝐶 Nat 𝐷) ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  wral 3061  {crab 3436  Vcvv 3480  [wsbc 3788  csb 3899  wss 3951  𝒫 cpw 4600  cop 4632   cuni 4907   × cxp 5683  ran crn 5686  wf 6557  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  m cmap 8866  Xcixp 8937  WUnicwun 10740  ndxcnx 17230  Basecbs 17247  Hom chom 17308  compcco 17309   Func cfunc 17899   Nat cnat 17989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-wun 10742  df-pnf 11297  df-mnf 11298  df-ltxr 11300  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-dec 12734  df-slot 17219  df-ndx 17231  df-base 17248  df-hom 17321  df-func 17903  df-nat 17991
This theorem is referenced by:  catcfuccl  18163
  Copyright terms: Public domain W3C validator