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Theorem wunnat 16387
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 16330 . . 3 (𝜑 → (𝐶 Func 𝐷) ∈ 𝑈)
51, 4, 4wunxp 9402 . 2 (𝜑 → ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)) ∈ 𝑈)
6 df-hom 15741 . . . . . . 7 Hom = Slot 14
76, 1, 3wunstr 15662 . . . . . 6 (𝜑 → (Hom ‘𝐷) ∈ 𝑈)
81, 7wunrn 9407 . . . . 5 (𝜑 → ran (Hom ‘𝐷) ∈ 𝑈)
91, 8wununi 9384 . . . 4 (𝜑 ran (Hom ‘𝐷) ∈ 𝑈)
10 df-base 15648 . . . . 5 Base = Slot 1
1110, 1, 2wunstr 15662 . . . 4 (𝜑 → (Base‘𝐶) ∈ 𝑈)
121, 9, 11wunmap 9404 . . 3 (𝜑 → ( ran (Hom ‘𝐷) ↑𝑚 (Base‘𝐶)) ∈ 𝑈)
131, 12wunpw 9385 . 2 (𝜑 → 𝒫 ( ran (Hom ‘𝐷) ↑𝑚 (Base‘𝐶)) ∈ 𝑈)
14 fvex 6097 . . . . . 6 (1st𝑓) ∈ V
15 fvex 6097 . . . . . . . . 9 (1st𝑔) ∈ V
16 ssrab2 3649 . . . . . . . . . . . . 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))
17 ovssunirn 6556 . . . . . . . . . . . . . . . 16 ((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ⊆ ran (Hom ‘𝐷)
1817rgenw 2907 . . . . . . . . . . . . . . 15 𝑥 ∈ (Base‘𝐶)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ⊆ ran (Hom ‘𝐷)
19 ss2ixp 7784 . . . . . . . . . . . . . . 15 (∀𝑥 ∈ (Base‘𝐶)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ⊆ ran (Hom ‘𝐷) → X𝑥 ∈ (Base‘𝐶)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ⊆ X𝑥 ∈ (Base‘𝐶) ran (Hom ‘𝐷))
2018, 19ax-mp 5 . . . . . . . . . . . . . 14 X𝑥 ∈ (Base‘𝐶)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ⊆ X𝑥 ∈ (Base‘𝐶) ran (Hom ‘𝐷)
21 fvex 6097 . . . . . . . . . . . . . . 15 (Base‘𝐶) ∈ V
22 fvex 6097 . . . . . . . . . . . . . . . . 17 (Hom ‘𝐷) ∈ V
2322rnex 6969 . . . . . . . . . . . . . . . 16 ran (Hom ‘𝐷) ∈ V
2423uniex 6828 . . . . . . . . . . . . . . 15 ran (Hom ‘𝐷) ∈ V
2521, 24ixpconst 7781 . . . . . . . . . . . . . 14 X𝑥 ∈ (Base‘𝐶) ran (Hom ‘𝐷) = ( ran (Hom ‘𝐷) ↑𝑚 (Base‘𝐶))
2620, 25sseqtri 3599 . . . . . . . . . . . . 13 X𝑥 ∈ (Base‘𝐶)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ⊆ ( ran (Hom ‘𝐷) ↑𝑚 (Base‘𝐶))
2716, 26sstri 3576 . . . . . . . . . . . 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))
28 ovex 6554 . . . . . . . . . . . . 13 ( ran (Hom ‘𝐷) ↑𝑚 (Base‘𝐶)) ∈ V
2928elpw2 4749 . . . . . . . . . . . 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))} ∈ 𝒫 ( ran (Hom ‘𝐷) ↑𝑚 (Base‘𝐶)) ↔ {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)))
3027, 29mpbir 219 . . . . . . . . . . 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))
3130sbcth 3416 . . . . . . . . . 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)))
32 sbcel1g 3938 . . . . . . . . . 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))))
3331, 32mpbid 220 . . . . . . . . 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)))
3415, 33ax-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))
3534sbcth 3416 . . . . . . 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)))
36 sbcel1g 3938 . . . . . . 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))))
3735, 36mpbid 220 . . . . . 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)))
3814, 37ax-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))
3938rgen2w 2908 . . . 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))
40 eqid 2609 . . . . . 6 (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
41 eqid 2609 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
42 eqid 2609 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
43 eqid 2609 . . . . . 6 (Hom ‘𝐷) = (Hom ‘𝐷)
44 eqid 2609 . . . . . 6 (comp‘𝐷) = (comp‘𝐷)
4540, 41, 42, 43, 44natfval 16377 . . . . 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))})
4645fmpt2 7103 . . . 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)))
4739, 46mpbi 218 . . 3 (𝐶 Nat 𝐷):((𝐶 Func 𝐷) × (𝐶 Func 𝐷))⟶𝒫 ( ran (Hom ‘𝐷) ↑𝑚 (Base‘𝐶))
4847a1i 11 . 2 (𝜑 → (𝐶 Nat 𝐷):((𝐶 Func 𝐷) × (𝐶 Func 𝐷))⟶𝒫 ( ran (Hom ‘𝐷) ↑𝑚 (Base‘𝐶)))
491, 5, 13, 48wunf 9405 1 (𝜑 → (𝐶 Nat 𝐷) ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  wral 2895  {crab 2899  Vcvv 3172  [wsbc 3401  csb 3498  wss 3539  𝒫 cpw 4107  cop 4130   cuni 4366   × cxp 5025  ran crn 5028  wf 5785  cfv 5789  (class class class)co 6526  1st c1st 7034  2nd c2nd 7035  𝑚 cmap 7721  Xcixp 7771  WUnicwun 9378  1c1 9793  4c4 10921  cdc 11327  Basecbs 15643  Hom chom 15727  compcco 15728   Func cfunc 16285   Nat cnat 16372
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-1st 7036  df-2nd 7037  df-map 7723  df-pm 7724  df-ixp 7772  df-wun 9380  df-slot 15647  df-base 15648  df-hom 15741  df-func 16289  df-nat 16374
This theorem is referenced by:  catcfuccl  16530
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