Theorem isnvlem 30703

Description: Lemma for isnv 30705. (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 30685	. . 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 2833	. 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 4807	. . . . 5 ⊢ (𝑔 = 𝐺 → ⟨𝑔, 𝑠⟩ = ⟨𝐺, 𝑠⟩)
4	3	eleq1d 2826	. . . 4 ⊢ (𝑔 = 𝐺 → (⟨𝑔, 𝑠⟩ ∈ CVecOLD ↔ ⟨𝐺, 𝑠⟩ ∈ CVecOLD))
5		rneq 5885	. . . . . 6 ⊢ (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
6		isnvlem.1	. . . . . 6 ⊢ 𝑋 = ran 𝐺
7	5, 6	eqtr4di 2794	. . . . 5 ⊢ (𝑔 = 𝐺 → ran 𝑔 = 𝑋)
8	7	feq2d 6643	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑛:ran 𝑔⟶ℝ ↔ 𝑛:𝑋⟶ℝ))
9		fveq2 6831	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (GId‘𝑔) = (GId‘𝐺))
10		isnvlem.2	. . . . . . . . 9 ⊢ 𝑍 = (GId‘𝐺)
11	9, 10	eqtr4di 2794	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (GId‘𝑔) = 𝑍)
12	11	eqeq2d 2752	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑥 = (GId‘𝑔) ↔ 𝑥 = 𝑍))
13	12	imbi2d 342	. . . . . 6 ⊢ (𝑔 = 𝐺 → (((𝑛‘𝑥) = 0 → 𝑥 = (GId‘𝑔)) ↔ ((𝑛‘𝑥) = 0 → 𝑥 = 𝑍)))
14		oveq 7366	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
15	14	fveq2d 6835	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑛‘(𝑥𝑔𝑦)) = (𝑛‘(𝑥𝐺𝑦)))
16	15	breq1d 5085	. . . . . . 7 ⊢ (𝑔 = 𝐺 → ((𝑛‘(𝑥𝑔𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦)) ↔ (𝑛‘(𝑥𝐺𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦))))
17	7, 16	raleqbidv 3315	. . . . . 6 ⊢ (𝑔 = 𝐺 → (∀𝑦 ∈ ran 𝑔(𝑛‘(𝑥𝑔𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦)) ↔ ∀𝑦 ∈ 𝑋 (𝑛‘(𝑥𝐺𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦))))
18	13, 17	3anbi13d 1447	. . . . 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 3315	. . . 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 1445	. . 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 4808	. . . . 5 ⊢ (𝑠 = 𝑆 → ⟨𝐺, 𝑠⟩ = ⟨𝐺, 𝑆⟩)
22	21	eleq1d 2826	. . . 4 ⊢ (𝑠 = 𝑆 → (⟨𝐺, 𝑠⟩ ∈ CVecOLD ↔ ⟨𝐺, 𝑆⟩ ∈ CVecOLD))
23		oveq 7366	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (𝑦𝑠𝑥) = (𝑦𝑆𝑥))
24	23	fveqeq2d 6839	. . . . . . 7 ⊢ (𝑠 = 𝑆 → ((𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ↔ (𝑛‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥))))
25	24	ralbidv 3164	. . . . . 6 ⊢ (𝑠 = 𝑆 → (∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ↔ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥))))
26	25	3anbi2d 1450	. . . . 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 3164	. . . 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 1447	. . 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 6637	. . . 4 ⊢ (𝑛 = 𝑁 → (𝑛:𝑋⟶ℝ ↔ 𝑁:𝑋⟶ℝ))
30		fveq1 6830	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑛‘𝑥) = (𝑁‘𝑥))
31	30	eqeq1d 2743	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝑛‘𝑥) = 0 ↔ (𝑁‘𝑥) = 0))
32	31	imbi1d 343	. . . . . 6 ⊢ (𝑛 = 𝑁 → (((𝑛‘𝑥) = 0 → 𝑥 = 𝑍) ↔ ((𝑁‘𝑥) = 0 → 𝑥 = 𝑍)))
33		fveq1 6830	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑛‘(𝑦𝑆𝑥)) = (𝑁‘(𝑦𝑆𝑥)))
34	30	oveq2d 7376	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → ((abs‘𝑦) · (𝑛‘𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)))
35	33, 34	eqeq12d 2757	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝑛‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ↔ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥))))
36	35	ralbidv 3164	. . . . . 6 ⊢ (𝑛 = 𝑁 → (∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ↔ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥))))
37		fveq1 6830	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑛‘(𝑥𝐺𝑦)) = (𝑁‘(𝑥𝐺𝑦)))
38		fveq1 6830	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (𝑛‘𝑦) = (𝑁‘𝑦))
39	30, 38	oveq12d 7378	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → ((𝑛‘𝑥) + (𝑛‘𝑦)) = ((𝑁‘𝑥) + (𝑁‘𝑦)))
40	37, 39	breq12d 5088	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝑛‘(𝑥𝐺𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦)) ↔ (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))
41	40	ralbidv 3164	. . . . . 6 ⊢ (𝑛 = 𝑁 → (∀𝑦 ∈ 𝑋 (𝑛‘(𝑥𝐺𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦)) ↔ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))
42	32, 36, 41	3anbi123d 1445	. . . . 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 3164	. . . 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 1448	. . 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 7470	. 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 285	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 208   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121  ∀wral 3055  Vcvv 3433  ⟨cop 4564   class class class wbr 5075  ran crn 5622  ⟶wf 6485  ‘cfv 6489  (class class class)co 7360  {coprab 7361  ℂcc 11031  ℝcr 11032  0cc0 11033   + caddc 11036   · cmul 11038   ≤ cle 11175  abscabs 15191  GIdcgi 30583  CVec_OLDcvc 30651  NrmCVeccnv 30677

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 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-fv 6497  df-ov 7363  df-oprab 7364  df-nv 30685

This theorem is referenced by:  isnv  30705