Theorem isnvlem 30332

Description: Lemma for isnv 30334. (Contributed by NM, 11-Nov-2006.) (New usage is discouraged.)

Hypotheses

Ref	Expression
isnvlem.1	⊢ 𝑋 = ran 𝐺
isnvlem.2	⊢ 𝑍 = (GId‘𝐺)

Assertion

Ref	Expression
isnvlem	⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V) → (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ↔ (⟨𝐺, 𝑆⟩ ∈ CVecOLD ∧ 𝑁:𝑋⟶ℝ ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))))

Distinct variable groups:   𝑥,𝑦,𝐺   𝑥,𝑁,𝑦   𝑥,𝑆,𝑦   𝑥,𝑋,𝑦

Allowed substitution hints:   𝑍(𝑥,𝑦)

Proof of Theorem isnvlem

Dummy variables 𝑔 𝑛 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nv 30314	. . 3 ⊢ NrmCVec = {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ (⟨𝑔, 𝑠⟩ ∈ CVecOLD ∧ 𝑛:ran 𝑔⟶ℝ ∧ ∀𝑥 ∈ ran 𝑔(((𝑛‘𝑥) = 0 → 𝑥 = (GId‘𝑔)) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ ran 𝑔(𝑛‘(𝑥𝑔𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦))))}
2	1	eleq2i 2817	. 2 ⊢ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ↔ ⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ (⟨𝑔, 𝑠⟩ ∈ CVecOLD ∧ 𝑛:ran 𝑔⟶ℝ ∧ ∀𝑥 ∈ ran 𝑔(((𝑛‘𝑥) = 0 → 𝑥 = (GId‘𝑔)) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ ran 𝑔(𝑛‘(𝑥𝑔𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦))))})
3		opeq1 4865	. . . . 5 ⊢ (𝑔 = 𝐺 → ⟨𝑔, 𝑠⟩ = ⟨𝐺, 𝑠⟩)
4	3	eleq1d 2810	. . . 4 ⊢ (𝑔 = 𝐺 → (⟨𝑔, 𝑠⟩ ∈ CVecOLD ↔ ⟨𝐺, 𝑠⟩ ∈ CVecOLD))
5		rneq 5925	. . . . . 6 ⊢ (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
6		isnvlem.1	. . . . . 6 ⊢ 𝑋 = ran 𝐺
7	5, 6	eqtr4di 2782	. . . . 5 ⊢ (𝑔 = 𝐺 → ran 𝑔 = 𝑋)
8	7	feq2d 6693	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑛:ran 𝑔⟶ℝ ↔ 𝑛:𝑋⟶ℝ))
9		fveq2 6881	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (GId‘𝑔) = (GId‘𝐺))
10		isnvlem.2	. . . . . . . . 9 ⊢ 𝑍 = (GId‘𝐺)
11	9, 10	eqtr4di 2782	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (GId‘𝑔) = 𝑍)
12	11	eqeq2d 2735	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑥 = (GId‘𝑔) ↔ 𝑥 = 𝑍))
13	12	imbi2d 340	. . . . . 6 ⊢ (𝑔 = 𝐺 → (((𝑛‘𝑥) = 0 → 𝑥 = (GId‘𝑔)) ↔ ((𝑛‘𝑥) = 0 → 𝑥 = 𝑍)))
14		oveq 7407	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
15	14	fveq2d 6885	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑛‘(𝑥𝑔𝑦)) = (𝑛‘(𝑥𝐺𝑦)))
16	15	breq1d 5148	. . . . . . 7 ⊢ (𝑔 = 𝐺 → ((𝑛‘(𝑥𝑔𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦)) ↔ (𝑛‘(𝑥𝐺𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦))))
17	7, 16	raleqbidv 3334	. . . . . 6 ⊢ (𝑔 = 𝐺 → (∀𝑦 ∈ ran 𝑔(𝑛‘(𝑥𝑔𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦)) ↔ ∀𝑦 ∈ 𝑋 (𝑛‘(𝑥𝐺𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦))))
18	13, 17	3anbi13d 1434	. . . . 5 ⊢ (𝑔 = 𝐺 → ((((𝑛‘𝑥) = 0 → 𝑥 = (GId‘𝑔)) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ ran 𝑔(𝑛‘(𝑥𝑔𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦))) ↔ (((𝑛‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑛‘(𝑥𝐺𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦)))))
19	7, 18	raleqbidv 3334	. . . 4 ⊢ (𝑔 = 𝐺 → (∀𝑥 ∈ ran 𝑔(((𝑛‘𝑥) = 0 → 𝑥 = (GId‘𝑔)) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ ran 𝑔(𝑛‘(𝑥𝑔𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦))) ↔ ∀𝑥 ∈ 𝑋 (((𝑛‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑛‘(𝑥𝐺𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦)))))
20	4, 8, 19	3anbi123d 1432	. . 3 ⊢ (𝑔 = 𝐺 → ((⟨𝑔, 𝑠⟩ ∈ CVecOLD ∧ 𝑛:ran 𝑔⟶ℝ ∧ ∀𝑥 ∈ ran 𝑔(((𝑛‘𝑥) = 0 → 𝑥 = (GId‘𝑔)) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ ran 𝑔(𝑛‘(𝑥𝑔𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦)))) ↔ (⟨𝐺, 𝑠⟩ ∈ CVecOLD ∧ 𝑛:𝑋⟶ℝ ∧ ∀𝑥 ∈ 𝑋 (((𝑛‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑛‘(𝑥𝐺𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦))))))
21		opeq2 4866	. . . . 5 ⊢ (𝑠 = 𝑆 → ⟨𝐺, 𝑠⟩ = ⟨𝐺, 𝑆⟩)
22	21	eleq1d 2810	. . . 4 ⊢ (𝑠 = 𝑆 → (⟨𝐺, 𝑠⟩ ∈ CVecOLD ↔ ⟨𝐺, 𝑆⟩ ∈ CVecOLD))
23		oveq 7407	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (𝑦𝑠𝑥) = (𝑦𝑆𝑥))
24	23	fveqeq2d 6889	. . . . . . 7 ⊢ (𝑠 = 𝑆 → ((𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ↔ (𝑛‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥))))
25	24	ralbidv 3169	. . . . . 6 ⊢ (𝑠 = 𝑆 → (∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ↔ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥))))
26	25	3anbi2d 1437	. . . . 5 ⊢ (𝑠 = 𝑆 → ((((𝑛‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑛‘(𝑥𝐺𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦))) ↔ (((𝑛‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑛‘(𝑥𝐺𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦)))))
27	26	ralbidv 3169	. . . 4 ⊢ (𝑠 = 𝑆 → (∀𝑥 ∈ 𝑋 (((𝑛‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑛‘(𝑥𝐺𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦))) ↔ ∀𝑥 ∈ 𝑋 (((𝑛‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑛‘(𝑥𝐺𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦)))))
28	22, 27	3anbi13d 1434	. . 3 ⊢ (𝑠 = 𝑆 → ((⟨𝐺, 𝑠⟩ ∈ CVecOLD ∧ 𝑛:𝑋⟶ℝ ∧ ∀𝑥 ∈ 𝑋 (((𝑛‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑛‘(𝑥𝐺𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦)))) ↔ (⟨𝐺, 𝑆⟩ ∈ CVecOLD ∧ 𝑛:𝑋⟶ℝ ∧ ∀𝑥 ∈ 𝑋 (((𝑛‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑛‘(𝑥𝐺𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦))))))
29		feq1 6688	. . . 4 ⊢ (𝑛 = 𝑁 → (𝑛:𝑋⟶ℝ ↔ 𝑁:𝑋⟶ℝ))
30		fveq1 6880	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑛‘𝑥) = (𝑁‘𝑥))
31	30	eqeq1d 2726	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝑛‘𝑥) = 0 ↔ (𝑁‘𝑥) = 0))
32	31	imbi1d 341	. . . . . 6 ⊢ (𝑛 = 𝑁 → (((𝑛‘𝑥) = 0 → 𝑥 = 𝑍) ↔ ((𝑁‘𝑥) = 0 → 𝑥 = 𝑍)))
33		fveq1 6880	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑛‘(𝑦𝑆𝑥)) = (𝑁‘(𝑦𝑆𝑥)))
34	30	oveq2d 7417	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → ((abs‘𝑦) · (𝑛‘𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)))
35	33, 34	eqeq12d 2740	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝑛‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ↔ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥))))
36	35	ralbidv 3169	. . . . . 6 ⊢ (𝑛 = 𝑁 → (∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ↔ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥))))
37		fveq1 6880	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑛‘(𝑥𝐺𝑦)) = (𝑁‘(𝑥𝐺𝑦)))
38		fveq1 6880	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (𝑛‘𝑦) = (𝑁‘𝑦))
39	30, 38	oveq12d 7419	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → ((𝑛‘𝑥) + (𝑛‘𝑦)) = ((𝑁‘𝑥) + (𝑁‘𝑦)))
40	37, 39	breq12d 5151	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝑛‘(𝑥𝐺𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦)) ↔ (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))
41	40	ralbidv 3169	. . . . . 6 ⊢ (𝑛 = 𝑁 → (∀𝑦 ∈ 𝑋 (𝑛‘(𝑥𝐺𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦)) ↔ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))
42	32, 36, 41	3anbi123d 1432	. . . . 5 ⊢ (𝑛 = 𝑁 → ((((𝑛‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑛‘(𝑥𝐺𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦))) ↔ (((𝑁‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))))
43	42	ralbidv 3169	. . . 4 ⊢ (𝑛 = 𝑁 → (∀𝑥 ∈ 𝑋 (((𝑛‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑛‘(𝑥𝐺𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦))) ↔ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))))
44	29, 43	3anbi23d 1435	. . 3 ⊢ (𝑛 = 𝑁 → ((⟨𝐺, 𝑆⟩ ∈ CVecOLD ∧ 𝑛:𝑋⟶ℝ ∧ ∀𝑥 ∈ 𝑋 (((𝑛‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑛‘(𝑥𝐺𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦)))) ↔ (⟨𝐺, 𝑆⟩ ∈ CVecOLD ∧ 𝑁:𝑋⟶ℝ ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))))
45	20, 28, 44	eloprabg 7510	. 2 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V) → (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ (⟨𝑔, 𝑠⟩ ∈ CVecOLD ∧ 𝑛:ran 𝑔⟶ℝ ∧ ∀𝑥 ∈ ran 𝑔(((𝑛‘𝑥) = 0 → 𝑥 = (GId‘𝑔)) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ ran 𝑔(𝑛‘(𝑥𝑔𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦))))} ↔ (⟨𝐺, 𝑆⟩ ∈ CVecOLD ∧ 𝑁:𝑋⟶ℝ ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))))
46	2, 45	bitrid 283	1 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V) → (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ↔ (⟨𝐺, 𝑆⟩ ∈ CVecOLD ∧ 𝑁:𝑋⟶ℝ ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∀wral 3053  Vcvv 3466  ⟨cop 4626   class class class wbr 5138  ran crn 5667  ⟶wf 6529  ‘cfv 6533  (class class class)co 7401  {coprab 7402  ℂcc 11104  ℝcr 11105  0cc0 11106   + caddc 11109   · cmul 11111   ≤ cle 11246  abscabs 15178  GIdcgi 30212  CVec_OLDcvc 30280  NrmCVeccnv 30306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7404  df-oprab 7405  df-nv 30314

This theorem is referenced by:  isnv  30334