Theorem wunnat 17881

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	⊢ (𝜑 → 𝑈 ∈ WUni)
wunnat.2	⊢ (𝜑 → 𝐶 ∈ 𝑈)
wunnat.3	⊢ (𝜑 → 𝐷 ∈ 𝑈)

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.

Step	Hyp	Ref	Expression
1		wunnat.1	. 2 ⊢ (𝜑 → 𝑈 ∈ WUni)
2		wunnat.2	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑈)
3		wunnat.3	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑈)
4	1, 2, 3	wunfunc 17823	. . 3 ⊢ (𝜑 → (𝐶 Func 𝐷) ∈ 𝑈)
5	1, 4, 4	wunxp 10633	. 2 ⊢ (𝜑 → ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)) ∈ 𝑈)
6		homid 17330	. . . . . . 7 ⊢ Hom = Slot (Hom ‘ndx)
7	6, 1, 3	wunstr 17113	. . . . . 6 ⊢ (𝜑 → (Hom ‘𝐷) ∈ 𝑈)
8	1, 7	wunrn 10638	. . . . 5 ⊢ (𝜑 → ran (Hom ‘𝐷) ∈ 𝑈)
9	1, 8	wununi 10615	. . . 4 ⊢ (𝜑 → ∪ ran (Hom ‘𝐷) ∈ 𝑈)
10		baseid 17137	. . . . 5 ⊢ Base = Slot (Base‘ndx)
11	10, 1, 2	wunstr 17113	. . . 4 ⊢ (𝜑 → (Base‘𝐶) ∈ 𝑈)
12	1, 9, 11	wunmap 10635	. . 3 ⊢ (𝜑 → (∪ ran (Hom ‘𝐷) ↑m (Base‘𝐶)) ∈ 𝑈)
13	1, 12	wunpw 10616	. 2 ⊢ (𝜑 → 𝒫 (∪ ran (Hom ‘𝐷) ↑m (Base‘𝐶)) ∈ 𝑈)
14		fvex 6845	. . . . . 6 ⊢ (1st ‘𝑓) ∈ V
15		fvex 6845	. . . . . . . . 9 ⊢ (1st ‘𝑔) ∈ V
16		ovex 7389	. . . . . . . . . . . 12 ⊢ (∪ ran (Hom ‘𝐷) ↑m (Base‘𝐶)) ∈ V
17		ssrab2 4030	. . . . . . . . . . . . 13 ⊢ {𝑎 ∈ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ⊆ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥))
18		ovssunirn 7392	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ⊆ ∪ ran (Hom ‘𝐷)
19	18	rgenw 3053	. . . . . . . . . . . . . . 15 ⊢ ∀𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ⊆ ∪ ran (Hom ‘𝐷)
20		ss2ixp 8846	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ⊆ ∪ ran (Hom ‘𝐷) → X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ⊆ X𝑥 ∈ (Base‘𝐶)∪ ran (Hom ‘𝐷))
21	19, 20	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ⊆ X𝑥 ∈ (Base‘𝐶)∪ ran (Hom ‘𝐷)
22		fvex 6845	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝐶) ∈ V
23		fvex 6845	. . . . . . . . . . . . . . . . 17 ⊢ (Hom ‘𝐷) ∈ V
24	23	rnex 7850	. . . . . . . . . . . . . . . 16 ⊢ ran (Hom ‘𝐷) ∈ V
25	24	uniex 7684	. . . . . . . . . . . . . . 15 ⊢ ∪ ran (Hom ‘𝐷) ∈ V
26	22, 25	ixpconst 8843	. . . . . . . . . . . . . 14 ⊢ X𝑥 ∈ (Base‘𝐶)∪ ran (Hom ‘𝐷) = (∪ ran (Hom ‘𝐷) ↑m (Base‘𝐶))
27	21, 26	sseqtri 3980	. . . . . . . . . . . . 13 ⊢ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ⊆ (∪ ran (Hom ‘𝐷) ↑m (Base‘𝐶))
28	17, 27	sstri 3941	. . . . . . . . . . . 12 ⊢ {𝑎 ∈ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ⊆ (∪ ran (Hom ‘𝐷) ↑m (Base‘𝐶))
29	16, 28	elpwi2 5278	. . . . . . . . . . 11 ⊢ {𝑎 ∈ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ∈ 𝒫 (∪ ran (Hom ‘𝐷) ↑m (Base‘𝐶))
30	29	sbcth 3753	. . . . . . . . . 10 ⊢ ((1st ‘𝑔) ∈ V → [(1st ‘𝑔) / 𝑠]{𝑎 ∈ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ∈ 𝒫 (∪ ran (Hom ‘𝐷) ↑m (Base‘𝐶)))
31		sbcel1g 4366	. . . . . . . . . 10 ⊢ ((1st ‘𝑔) ∈ V → ([(1st ‘𝑔) / 𝑠]{𝑎 ∈ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ∈ 𝒫 (∪ ran (Hom ‘𝐷) ↑m (Base‘𝐶)) ↔ ⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ∈ 𝒫 (∪ ran (Hom ‘𝐷) ↑m (Base‘𝐶))))
32	30, 31	mpbid 232	. . . . . . . . 9 ⊢ ((1st ‘𝑔) ∈ V → ⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ∈ 𝒫 (∪ ran (Hom ‘𝐷) ↑m (Base‘𝐶)))
33	15, 32	ax-mp 5	. . . . . . . 8 ⊢ ⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ∈ 𝒫 (∪ ran (Hom ‘𝐷) ↑m (Base‘𝐶))
34	33	sbcth 3753	. . . . . . 7 ⊢ ((1st ‘𝑓) ∈ V → [(1st ‘𝑓) / 𝑟]⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ∈ 𝒫 (∪ ran (Hom ‘𝐷) ↑m (Base‘𝐶)))
35		sbcel1g 4366	. . . . . . 7 ⊢ ((1st ‘𝑓) ∈ V → ([(1st ‘𝑓) / 𝑟]⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ∈ 𝒫 (∪ ran (Hom ‘𝐷) ↑m (Base‘𝐶)) ↔ ⦋(1st ‘𝑓) / 𝑟⦌⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ∈ 𝒫 (∪ ran (Hom ‘𝐷) ↑m (Base‘𝐶))))
36	34, 35	mpbid 232	. . . . . 6 ⊢ ((1st ‘𝑓) ∈ V → ⦋(1st ‘𝑓) / 𝑟⦌⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ∈ 𝒫 (∪ ran (Hom ‘𝐷) ↑m (Base‘𝐶)))
37	14, 36	ax-mp 5	. . . . 5 ⊢ ⦋(1st ‘𝑓) / 𝑟⦌⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ∈ 𝒫 (∪ ran (Hom ‘𝐷) ↑m (Base‘𝐶))
38	37	rgen2w 3054	. . . 4 ⊢ ∀𝑓 ∈ (𝐶 Func 𝐷)∀𝑔 ∈ (𝐶 Func 𝐷)⦋(1st ‘𝑓) / 𝑟⦌⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ∈ 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)
44	39, 40, 41, 42, 43	natfval 17871	. . . . 5 ⊢ (𝐶 Nat 𝐷) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑔 ∈ (𝐶 Func 𝐷) ↦ ⦋(1st ‘𝑓) / 𝑟⦌⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))})
45	44	fmpo 8010	. . . 4 ⊢ (∀𝑓 ∈ (𝐶 Func 𝐷)∀𝑔 ∈ (𝐶 Func 𝐷)⦋(1st ‘𝑓) / 𝑟⦌⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ∈ 𝒫 (∪ ran (Hom ‘𝐷) ↑m (Base‘𝐶)) ↔ (𝐶 Nat 𝐷):((𝐶 Func 𝐷) × (𝐶 Func 𝐷))⟶𝒫 (∪ ran (Hom ‘𝐷) ↑m (Base‘𝐶)))
46	38, 45	mpbi 230	. . 3 ⊢ (𝐶 Nat 𝐷):((𝐶 Func 𝐷) × (𝐶 Func 𝐷))⟶𝒫 (∪ ran (Hom ‘𝐷) ↑m (Base‘𝐶))
47	46	a1i 11	. 2 ⊢ (𝜑 → (𝐶 Nat 𝐷):((𝐶 Func 𝐷) × (𝐶 Func 𝐷))⟶𝒫 (∪ ran (Hom ‘𝐷) ↑m (Base‘𝐶)))
48	1, 5, 13, 47	wunf 10636	1 ⊢ (𝜑 → (𝐶 Nat 𝐷) ∈ 𝑈)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  ∀wral 3049  {crab 3397  Vcvv 3438  [wsbc 3738  ⦋csb 3847   ⊆ wss 3899  𝒫 cpw 4552  ⟨cop 4584  ∪ cuni 4861   × cxp 5620  ran crn 5623  ⟶wf 6486  ‘cfv 6490  (class class class)co 7356  1^st c1st 7929  2^nd c2nd 7930   ↑_m cmap 8761  Xcixp 8833  WUnicwun 10609  ndxcnx 17118  Basecbs 17134  Hom chom 17186  compcco 17187   Func cfunc 17776   Nat cnat 17866

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-er 8633  df-map 8763  df-pm 8764  df-ixp 8834  df-en 8882  df-dom 8883  df-sdom 8884  df-wun 10611  df-pnf 11166  df-mnf 11167  df-ltxr 11169  df-nn 12144  df-2 12206  df-3 12207  df-4 12208  df-5 12209  df-6 12210  df-7 12211  df-8 12212  df-9 12213  df-n0 12400  df-dec 12606  df-slot 17107  df-ndx 17119  df-base 17135  df-hom 17199  df-func 17780  df-nat 17868

This theorem is referenced by:  catcfuccl  18040