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Theorem wunnatOLD 17905

Description: Obsolete proof of wunnat 17904 as of 13-Oct-2024. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
wunnat.1	⊢ (𝜑 → 𝑈 ∈ WUni)
wunnat.2	⊢ (𝜑 → 𝐶 ∈ 𝑈)
wunnat.3	⊢ (𝜑 → 𝐷 ∈ 𝑈)

**Assertion**
Ref	Expression
wunnatOLD	⊢ (𝜑 → (𝐶 Nat 𝐷) ∈ 𝑈)

Proof of Theorem wunnatOLD 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 17846	. . 3 ⊢ (𝜑 → (𝐶 Func 𝐷) ∈ 𝑈)
5	1, 4, 4	wunxp 10716	. 2 ⊢ (𝜑 → ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)) ∈ 𝑈)
6		df-hom 17218	. . . . . . 7 ⊢ Hom = Slot ;14
7	6, 1, 3	wunstr 17118	. . . . . 6 ⊢ (𝜑 → (Hom ‘𝐷) ∈ 𝑈)
8	1, 7	wunrn 10721	. . . . 5 ⊢ (𝜑 → ran (Hom ‘𝐷) ∈ 𝑈)
9	1, 8	wununi 10698	. . . 4 ⊢ (𝜑 → ∪ ran (Hom ‘𝐷) ∈ 𝑈)
10		df-base 17142	. . . . 5 ⊢ Base = Slot 1
11	10, 1, 2	wunstr 17118	. . . 4 ⊢ (𝜑 → (Base‘𝐶) ∈ 𝑈)
12	1, 9, 11	wunmap 10718	. . 3 ⊢ (𝜑 → (∪ ran (Hom ‘𝐷) ↑_m (Base‘𝐶)) ∈ 𝑈)
13	1, 12	wunpw 10699	. 2 ⊢ (𝜑 → 𝒫 (∪ ran (Hom ‘𝐷) ↑_m (Base‘𝐶)) ∈ 𝑈)
14		fvex 6902	. . . . . 6 ⊢ (1^st ‘𝑓) ∈ V
15		fvex 6902	. . . . . . . . 9 ⊢ (1^st ‘𝑔) ∈ V
16		ovex 7439	. . . . . . . . . . . 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 7442	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ⊆ ∪ ran (Hom ‘𝐷)
19	18	rgenw 3066	. . . . . . . . . . . . . . 15 ⊢ ∀𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ⊆ ∪ ran (Hom ‘𝐷)
20		ss2ixp 8901	. . . . . . . . . . . . . . 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 6902	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝐶) ∈ V
23		fvex 6902	. . . . . . . . . . . . . . . . 17 ⊢ (Hom ‘𝐷) ∈ V
24	23	rnex 7900	. . . . . . . . . . . . . . . 16 ⊢ ran (Hom ‘𝐷) ∈ V
25	24	uniex 7728	. . . . . . . . . . . . . . 15 ⊢ ∪ ran (Hom ‘𝐷) ∈ V
26	22, 25	ixpconst 8898	. . . . . . . . . . . . . 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 3067	. . . 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 2733	. . . . . 6 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
40		eqid 2733	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
41		eqid 2733	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
42		eqid 2733	. . . . . 6 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
43		eqid 2733	. . . . . 6 ⊢ (comp‘𝐷) = (comp‘𝐷)
44	39, 40, 41, 42, 43	natfval 17894	. . . . 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 8051	. . . 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 10719	1 ⊢ (𝜑 → (𝐶 Nat 𝐷) ∈ 𝑈)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ∀wral 3062 {crab 3433 Vcvv 3475 [wsbc 3777 ⦋csb 3893 ⊆ wss 3948 𝒫 cpw 4602 ⟨cop 4634 ∪ cuni 4908 × cxp 5674 ran crn 5677 ⟶wf 6537 ‘cfv 6541 (class class class)co 7406 1^st c1st 7970 2^nd c2nd 7971 ↑_m cmap 8817 Xcixp 8888 WUnicwun 10692 1c1 11108 4c4 12266 cdc 12674 Basecbs 17141 Hom chom 17205 compcco 17206 Func cfunc 17801 Nat cnat 17889

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 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 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-er 8700 df-map 8819 df-pm 8820 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-wun 10694 df-pnf 11247 df-mnf 11248 df-ltxr 11250 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-dec 12675 df-slot 17112 df-ndx 17124 df-base 17142 df-hom 17218 df-func 17805 df-nat 17891

This theorem is referenced by: (None)

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