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Theorem wunnat 17917

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 17858	. . 3 ⊢ (𝜑 → (𝐶 Func 𝐷) ∈ 𝑈)
5	1, 4, 4	wunxp 10725	. 2 ⊢ (𝜑 → ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)) ∈ 𝑈)
6		homid 17364	. . . . . . 7 ⊢ Hom = Slot (Hom ‘ndx)
7	6, 1, 3	wunstr 17128	. . . . . 6 ⊢ (𝜑 → (Hom ‘𝐷) ∈ 𝑈)
8	1, 7	wunrn 10730	. . . . 5 ⊢ (𝜑 → ran (Hom ‘𝐷) ∈ 𝑈)
9	1, 8	wununi 10707	. . . 4 ⊢ (𝜑 → ∪ ran (Hom ‘𝐷) ∈ 𝑈)
10		baseid 17154	. . . . 5 ⊢ Base = Slot (Base‘ndx)
11	10, 1, 2	wunstr 17128	. . . 4 ⊢ (𝜑 → (Base‘𝐶) ∈ 𝑈)
12	1, 9, 11	wunmap 10727	. . 3 ⊢ (𝜑 → (∪ ran (Hom ‘𝐷) ↑_m (Base‘𝐶)) ∈ 𝑈)
13	1, 12	wunpw 10708	. 2 ⊢ (𝜑 → 𝒫 (∪ ran (Hom ‘𝐷) ↑_m (Base‘𝐶)) ∈ 𝑈)
14		fvex 6904	. . . . . 6 ⊢ (1^st ‘𝑓) ∈ V
15		fvex 6904	. . . . . . . . 9 ⊢ (1^st ‘𝑔) ∈ V
16		ovex 7445	. . . . . . . . . . . 12 ⊢ (∪ ran (Hom ‘𝐷) ↑_m (Base‘𝐶)) ∈ V
17		ssrab2 4077	. . . . . . . . . . . . 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(2^nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2^nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ⊆ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥))
18		ovssunirn 7448	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ⊆ ∪ ran (Hom ‘𝐷)
19	18	rgenw 3064	. . . . . . . . . . . . . . 15 ⊢ ∀𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ⊆ ∪ ran (Hom ‘𝐷)
20		ss2ixp 8910	. . . . . . . . . . . . . . 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 6904	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝐶) ∈ V
23		fvex 6904	. . . . . . . . . . . . . . . . 17 ⊢ (Hom ‘𝐷) ∈ V
24	23	rnex 7907	. . . . . . . . . . . . . . . 16 ⊢ ran (Hom ‘𝐷) ∈ V
25	24	uniex 7735	. . . . . . . . . . . . . . 15 ⊢ ∪ ran (Hom ‘𝐷) ∈ V
26	22, 25	ixpconst 8907	. . . . . . . . . . . . . 14 ⊢ X𝑥 ∈ (Base‘𝐶)∪ ran (Hom ‘𝐷) = (∪ ran (Hom ‘𝐷) ↑_m (Base‘𝐶))
27	21, 26	sseqtri 4018	. . . . . . . . . . . . 13 ⊢ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ⊆ (∪ ran (Hom ‘𝐷) ↑_m (Base‘𝐶))
28	17, 27	sstri 3991	. . . . . . . . . . . 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(2^nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2^nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ⊆ (∪ ran (Hom ‘𝐷) ↑_m (Base‘𝐶))
29	16, 28	elpwi2 5346	. . . . . . . . . . 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(2^nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2^nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ∈ 𝒫 (∪ ran (Hom ‘𝐷) ↑_m (Base‘𝐶))
30	29	sbcth 3792	. . . . . . . . . 10 ⊢ ((1^st ‘𝑔) ∈ V → [(1^st ‘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(2^nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2^nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ∈ 𝒫 (∪ ran (Hom ‘𝐷) ↑_m (Base‘𝐶)))
31		sbcel1g 4413	. . . . . . . . . 10 ⊢ ((1^st ‘𝑔) ∈ V → ([(1^st ‘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(2^nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2^nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ∈ 𝒫 (∪ ran (Hom ‘𝐷) ↑_m (Base‘𝐶)) ↔ ⦋(1^st ‘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(2^nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2^nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ∈ 𝒫 (∪ ran (Hom ‘𝐷) ↑_m (Base‘𝐶))))
32	30, 31	mpbid 231	. . . . . . . . 9 ⊢ ((1^st ‘𝑔) ∈ V → ⦋(1^st ‘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(2^nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2^nd ‘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 ⊢ ⦋(1^st ‘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(2^nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2^nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ∈ 𝒫 (∪ ran (Hom ‘𝐷) ↑_m (Base‘𝐶))
34	33	sbcth 3792	. . . . . . 7 ⊢ ((1^st ‘𝑓) ∈ V → [(1^st ‘𝑓) / 𝑟]⦋(1^st ‘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(2^nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2^nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ∈ 𝒫 (∪ ran (Hom ‘𝐷) ↑_m (Base‘𝐶)))
35		sbcel1g 4413	. . . . . . 7 ⊢ ((1^st ‘𝑓) ∈ V → ([(1^st ‘𝑓) / 𝑟]⦋(1^st ‘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(2^nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2^nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ∈ 𝒫 (∪ ran (Hom ‘𝐷) ↑_m (Base‘𝐶)) ↔ ⦋(1^st ‘𝑓) / 𝑟⦌⦋(1^st ‘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(2^nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2^nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ∈ 𝒫 (∪ ran (Hom ‘𝐷) ↑_m (Base‘𝐶))))
36	34, 35	mpbid 231	. . . . . 6 ⊢ ((1^st ‘𝑓) ∈ V → ⦋(1^st ‘𝑓) / 𝑟⦌⦋(1^st ‘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(2^nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2^nd ‘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 ⊢ ⦋(1^st ‘𝑓) / 𝑟⦌⦋(1^st ‘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(2^nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2^nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ∈ 𝒫 (∪ ran (Hom ‘𝐷) ↑_m (Base‘𝐶))
38	37	rgen2w 3065	. . . 4 ⊢ ∀𝑓 ∈ (𝐶 Func 𝐷)∀𝑔 ∈ (𝐶 Func 𝐷)⦋(1^st ‘𝑓) / 𝑟⦌⦋(1^st ‘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(2^nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2^nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ∈ 𝒫 (∪ ran (Hom ‘𝐷) ↑_m (Base‘𝐶))
39		eqid 2731	. . . . . 6 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
40		eqid 2731	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
41		eqid 2731	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
42		eqid 2731	. . . . . 6 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
43		eqid 2731	. . . . . 6 ⊢ (comp‘𝐷) = (comp‘𝐷)
44	39, 40, 41, 42, 43	natfval 17907	. . . . 5 ⊢ (𝐶 Nat 𝐷) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑔 ∈ (𝐶 Func 𝐷) ↦ ⦋(1^st ‘𝑓) / 𝑟⦌⦋(1^st ‘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(2^nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2^nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))})
45	44	fmpo 8058	. . . 4 ⊢ (∀𝑓 ∈ (𝐶 Func 𝐷)∀𝑔 ∈ (𝐶 Func 𝐷)⦋(1^st ‘𝑓) / 𝑟⦌⦋(1^st ‘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(2^nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2^nd ‘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 229	. . 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 10728	1 ⊢ (𝜑 → (𝐶 Nat 𝐷) ∈ 𝑈)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 ∀wral 3060 {crab 3431 Vcvv 3473 [wsbc 3777 ⦋csb 3893 ⊆ wss 3948 𝒫 cpw 4602 ⟨cop 4634 ∪ cuni 4908 × cxp 5674 ran crn 5677 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 1^st c1st 7977 2^nd c2nd 7978 ↑_m cmap 8826 Xcixp 8897 WUnicwun 10701 ndxcnx 17133 Basecbs 17151 Hom chom 17215 compcco 17216 Func cfunc 17811 Nat cnat 17902

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-map 8828 df-pm 8829 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-wun 10703 df-pnf 11257 df-mnf 11258 df-ltxr 11260 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-dec 12685 df-slot 17122 df-ndx 17134 df-base 17152 df-hom 17228 df-func 17815 df-nat 17904

This theorem is referenced by: catcfuccl 18079 catcfucclOLD 18080

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