Theorem wunnat 17463

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 17405	. . 3 ⊢ (𝜑 → (𝐶 Func 𝐷) ∈ 𝑈)
5	1, 4, 4	wunxp 10338	. 2 ⊢ (𝜑 → ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)) ∈ 𝑈)
6		homid 16919	. . . . . . 7 ⊢ Hom = Slot (Hom ‘ndx)
7	6, 1, 3	wunstr 16741	. . . . . 6 ⊢ (𝜑 → (Hom ‘𝐷) ∈ 𝑈)
8	1, 7	wunrn 10343	. . . . 5 ⊢ (𝜑 → ran (Hom ‘𝐷) ∈ 𝑈)
9	1, 8	wununi 10320	. . . 4 ⊢ (𝜑 → ∪ ran (Hom ‘𝐷) ∈ 𝑈)
10		baseid 16763	. . . . 5 ⊢ Base = Slot (Base‘ndx)
11	10, 1, 2	wunstr 16741	. . . 4 ⊢ (𝜑 → (Base‘𝐶) ∈ 𝑈)
12	1, 9, 11	wunmap 10340	. . 3 ⊢ (𝜑 → (∪ ran (Hom ‘𝐷) ↑m (Base‘𝐶)) ∈ 𝑈)
13	1, 12	wunpw 10321	. 2 ⊢ (𝜑 → 𝒫 (∪ ran (Hom ‘𝐷) ↑m (Base‘𝐶)) ∈ 𝑈)
14		fvex 6730	. . . . . 6 ⊢ (1st ‘𝑓) ∈ V
15		fvex 6730	. . . . . . . . 9 ⊢ (1st ‘𝑔) ∈ V
16		ovex 7246	. . . . . . . . . . . 12 ⊢ (∪ ran (Hom ‘𝐷) ↑m (Base‘𝐶)) ∈ V
17		ssrab2 3993	. . . . . . . . . . . . 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 7249	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ⊆ ∪ ran (Hom ‘𝐷)
19	18	rgenw 3073	. . . . . . . . . . . . . . 15 ⊢ ∀𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ⊆ ∪ ran (Hom ‘𝐷)
20		ss2ixp 8591	. . . . . . . . . . . . . . 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 6730	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝐶) ∈ V
23		fvex 6730	. . . . . . . . . . . . . . . . 17 ⊢ (Hom ‘𝐷) ∈ V
24	23	rnex 7690	. . . . . . . . . . . . . . . 16 ⊢ ran (Hom ‘𝐷) ∈ V
25	24	uniex 7529	. . . . . . . . . . . . . . 15 ⊢ ∪ ran (Hom ‘𝐷) ∈ V
26	22, 25	ixpconst 8588	. . . . . . . . . . . . . 14 ⊢ X𝑥 ∈ (Base‘𝐶)∪ ran (Hom ‘𝐷) = (∪ ran (Hom ‘𝐷) ↑m (Base‘𝐶))
27	21, 26	sseqtri 3937	. . . . . . . . . . . . 13 ⊢ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ⊆ (∪ ran (Hom ‘𝐷) ↑m (Base‘𝐶))
28	17, 27	sstri 3910	. . . . . . . . . . . 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 5239	. . . . . . . . . . 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 3709	. . . . . . . . . 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 4328	. . . . . . . . . 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 235	. . . . . . . . 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 3709	. . . . . . 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 4328	. . . . . . 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 235	. . . . . 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 3074	. . . 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 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)
44	39, 40, 41, 42, 43	natfval 17453	. . . . 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 7838	. . . 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 233	. . 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 10341	1 ⊢ (𝜑 → (𝐶 Nat 𝐷) ∈ 𝑈)

Syntax hints:   → wi 4   = wceq 1543   ∈ wcel 2110  ∀wral 3061  {crab 3065  Vcvv 3408  [wsbc 3694  ⦋csb 3811   ⊆ wss 3866  𝒫 cpw 4513  ⟨cop 4547  ∪ cuni 4819   × cxp 5549  ran crn 5552  ⟶wf 6376  ‘cfv 6380  (class class class)co 7213  1^st c1st 7759  2^nd c2nd 7760   ↑_m cmap 8508  Xcixp 8578  WUnicwun 10314  ndxcnx 16744  Basecbs 16760  Hom chom 16813  compcco 16814   Func cfunc 17360   Nat cnat 17448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-er 8391  df-map 8510  df-pm 8511  df-ixp 8579  df-en 8627  df-dom 8628  df-sdom 8629  df-wun 10316  df-pnf 10869  df-mnf 10870  df-ltxr 10872  df-nn 11831  df-2 11893  df-3 11894  df-4 11895  df-5 11896  df-6 11897  df-7 11898  df-8 11899  df-9 11900  df-n0 12091  df-dec 12294  df-slot 16735  df-ndx 16745  df-base 16761  df-hom 16826  df-func 17364  df-nat 17450

This theorem is referenced by:  catcfuccl  17625  catcfucclOLD  17626