Theorem iuninc 32573

Description: The union of an increasing collection of sets is its last element. (Contributed by Thierry Arnoux, 22-Jan-2017.)

Hypotheses

Ref	Expression
iuninc.1	⊢ (𝜑 → 𝐹 Fn ℕ)
iuninc.2	⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ⊆ (𝐹‘(𝑛 + 1)))

Assertion

Ref	Expression
iuninc	⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ∪ 𝑛 ∈ (1...𝑖)(𝐹‘𝑛) = (𝐹‘𝑖))

Distinct variable groups:   𝑖,𝑛   𝑛,𝐹   𝜑,𝑛

Allowed substitution hints:   𝜑(𝑖)   𝐹(𝑖)

Proof of Theorem iuninc

Dummy variables 𝑗 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7439	. . . . . 6 ⊢ (𝑗 = 1 → (1...𝑗) = (1...1))
2	1	iuneq1d 5019	. . . . 5 ⊢ (𝑗 = 1 → ∪ 𝑛 ∈ (1...𝑗)(𝐹‘𝑛) = ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛))
3		fveq2 6906	. . . . 5 ⊢ (𝑗 = 1 → (𝐹‘𝑗) = (𝐹‘1))
4	2, 3	eqeq12d 2753	. . . 4 ⊢ (𝑗 = 1 → (∪ 𝑛 ∈ (1...𝑗)(𝐹‘𝑛) = (𝐹‘𝑗) ↔ ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛) = (𝐹‘1)))
5	4	imbi2d 340	. . 3 ⊢ (𝑗 = 1 → ((𝜑 → ∪ 𝑛 ∈ (1...𝑗)(𝐹‘𝑛) = (𝐹‘𝑗)) ↔ (𝜑 → ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛) = (𝐹‘1))))
6		oveq2 7439	. . . . . 6 ⊢ (𝑗 = 𝑘 → (1...𝑗) = (1...𝑘))
7	6	iuneq1d 5019	. . . . 5 ⊢ (𝑗 = 𝑘 → ∪ 𝑛 ∈ (1...𝑗)(𝐹‘𝑛) = ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))
8		fveq2 6906	. . . . 5 ⊢ (𝑗 = 𝑘 → (𝐹‘𝑗) = (𝐹‘𝑘))
9	7, 8	eqeq12d 2753	. . . 4 ⊢ (𝑗 = 𝑘 → (∪ 𝑛 ∈ (1...𝑗)(𝐹‘𝑛) = (𝐹‘𝑗) ↔ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) = (𝐹‘𝑘)))
10	9	imbi2d 340	. . 3 ⊢ (𝑗 = 𝑘 → ((𝜑 → ∪ 𝑛 ∈ (1...𝑗)(𝐹‘𝑛) = (𝐹‘𝑗)) ↔ (𝜑 → ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) = (𝐹‘𝑘))))
11		oveq2 7439	. . . . . 6 ⊢ (𝑗 = (𝑘 + 1) → (1...𝑗) = (1...(𝑘 + 1)))
12	11	iuneq1d 5019	. . . . 5 ⊢ (𝑗 = (𝑘 + 1) → ∪ 𝑛 ∈ (1...𝑗)(𝐹‘𝑛) = ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))
13		fveq2 6906	. . . . 5 ⊢ (𝑗 = (𝑘 + 1) → (𝐹‘𝑗) = (𝐹‘(𝑘 + 1)))
14	12, 13	eqeq12d 2753	. . . 4 ⊢ (𝑗 = (𝑘 + 1) → (∪ 𝑛 ∈ (1...𝑗)(𝐹‘𝑛) = (𝐹‘𝑗) ↔ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) = (𝐹‘(𝑘 + 1))))
15	14	imbi2d 340	. . 3 ⊢ (𝑗 = (𝑘 + 1) → ((𝜑 → ∪ 𝑛 ∈ (1...𝑗)(𝐹‘𝑛) = (𝐹‘𝑗)) ↔ (𝜑 → ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) = (𝐹‘(𝑘 + 1)))))
16		oveq2 7439	. . . . . 6 ⊢ (𝑗 = 𝑖 → (1...𝑗) = (1...𝑖))
17	16	iuneq1d 5019	. . . . 5 ⊢ (𝑗 = 𝑖 → ∪ 𝑛 ∈ (1...𝑗)(𝐹‘𝑛) = ∪ 𝑛 ∈ (1...𝑖)(𝐹‘𝑛))
18		fveq2 6906	. . . . 5 ⊢ (𝑗 = 𝑖 → (𝐹‘𝑗) = (𝐹‘𝑖))
19	17, 18	eqeq12d 2753	. . . 4 ⊢ (𝑗 = 𝑖 → (∪ 𝑛 ∈ (1...𝑗)(𝐹‘𝑛) = (𝐹‘𝑗) ↔ ∪ 𝑛 ∈ (1...𝑖)(𝐹‘𝑛) = (𝐹‘𝑖)))
20	19	imbi2d 340	. . 3 ⊢ (𝑗 = 𝑖 → ((𝜑 → ∪ 𝑛 ∈ (1...𝑗)(𝐹‘𝑛) = (𝐹‘𝑗)) ↔ (𝜑 → ∪ 𝑛 ∈ (1...𝑖)(𝐹‘𝑛) = (𝐹‘𝑖))))
21		1z 12647	. . . . . 6 ⊢ 1 ∈ ℤ
22		fzsn 13606	. . . . . 6 ⊢ (1 ∈ ℤ → (1...1) = {1})
23		iuneq1 5008	. . . . . 6 ⊢ ((1...1) = {1} → ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛) = ∪ 𝑛 ∈ {1} (𝐹‘𝑛))
24	21, 22, 23	mp2b 10	. . . . 5 ⊢ ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛) = ∪ 𝑛 ∈ {1} (𝐹‘𝑛)
25		1ex 11257	. . . . . 6 ⊢ 1 ∈ V
26		fveq2 6906	. . . . . 6 ⊢ (𝑛 = 1 → (𝐹‘𝑛) = (𝐹‘1))
27	25, 26	iunxsn 5091	. . . . 5 ⊢ ∪ 𝑛 ∈ {1} (𝐹‘𝑛) = (𝐹‘1)
28	24, 27	eqtri 2765	. . . 4 ⊢ ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛) = (𝐹‘1)
29	28	a1i 11	. . 3 ⊢ (𝜑 → ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛) = (𝐹‘1))
30		simpll 767	. . . . . . 7 ⊢ (((𝑘 ∈ ℕ ∧ 𝜑) ∧ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) = (𝐹‘𝑘)) → 𝑘 ∈ ℕ)
31		elnnuz 12922	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ ↔ 𝑘 ∈ (ℤ≥‘1))
32		fzsuc 13611	. . . . . . . . . 10 ⊢ (𝑘 ∈ (ℤ≥‘1) → (1...(𝑘 + 1)) = ((1...𝑘) ∪ {(𝑘 + 1)}))
33	31, 32	sylbi 217	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (1...(𝑘 + 1)) = ((1...𝑘) ∪ {(𝑘 + 1)}))
34	33	iuneq1d 5019	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) = ∪ 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹‘𝑛))
35		iunxun 5094	. . . . . . . . 9 ⊢ ∪ 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹‘𝑛) = (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ ∪ 𝑛 ∈ {(𝑘 + 1)} (𝐹‘𝑛))
36		ovex 7464	. . . . . . . . . . 11 ⊢ (𝑘 + 1) ∈ V
37		fveq2 6906	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑘 + 1) → (𝐹‘𝑛) = (𝐹‘(𝑘 + 1)))
38	36, 37	iunxsn 5091	. . . . . . . . . 10 ⊢ ∪ 𝑛 ∈ {(𝑘 + 1)} (𝐹‘𝑛) = (𝐹‘(𝑘 + 1))
39	38	uneq2i 4165	. . . . . . . . 9 ⊢ (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ ∪ 𝑛 ∈ {(𝑘 + 1)} (𝐹‘𝑛)) = (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ (𝐹‘(𝑘 + 1)))
40	35, 39	eqtri 2765	. . . . . . . 8 ⊢ ∪ 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹‘𝑛) = (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ (𝐹‘(𝑘 + 1)))
41	34, 40	eqtrdi 2793	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) = (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ (𝐹‘(𝑘 + 1))))
42	30, 41	syl 17	. . . . . 6 ⊢ (((𝑘 ∈ ℕ ∧ 𝜑) ∧ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) = (𝐹‘𝑘)) → ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) = (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ (𝐹‘(𝑘 + 1))))
43		simpr 484	. . . . . . 7 ⊢ (((𝑘 ∈ ℕ ∧ 𝜑) ∧ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) = (𝐹‘𝑘)) → ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) = (𝐹‘𝑘))
44	43	uneq1d 4167	. . . . . 6 ⊢ (((𝑘 ∈ ℕ ∧ 𝜑) ∧ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) = (𝐹‘𝑘)) → (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ (𝐹‘(𝑘 + 1))) = ((𝐹‘𝑘) ∪ (𝐹‘(𝑘 + 1))))
45		simplr 769	. . . . . . 7 ⊢ (((𝑘 ∈ ℕ ∧ 𝜑) ∧ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) = (𝐹‘𝑘)) → 𝜑)
46		iuninc.2	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ⊆ (𝐹‘(𝑛 + 1)))
47	46	sbt 2066	. . . . . . . . 9 ⊢ [𝑘 / 𝑛]((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ⊆ (𝐹‘(𝑛 + 1)))
48		sbim 2303	. . . . . . . . . 10 ⊢ ([𝑘 / 𝑛]((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ⊆ (𝐹‘(𝑛 + 1))) ↔ ([𝑘 / 𝑛](𝜑 ∧ 𝑛 ∈ ℕ) → [𝑘 / 𝑛](𝐹‘𝑛) ⊆ (𝐹‘(𝑛 + 1))))
49		sban 2080	. . . . . . . . . . . 12 ⊢ ([𝑘 / 𝑛](𝜑 ∧ 𝑛 ∈ ℕ) ↔ ([𝑘 / 𝑛]𝜑 ∧ [𝑘 / 𝑛]𝑛 ∈ ℕ))
50		sbv 2088	. . . . . . . . . . . . 13 ⊢ ([𝑘 / 𝑛]𝜑 ↔ 𝜑)
51		clelsb1 2868	. . . . . . . . . . . . 13 ⊢ ([𝑘 / 𝑛]𝑛 ∈ ℕ ↔ 𝑘 ∈ ℕ)
52	50, 51	anbi12i 628	. . . . . . . . . . . 12 ⊢ (([𝑘 / 𝑛]𝜑 ∧ [𝑘 / 𝑛]𝑛 ∈ ℕ) ↔ (𝜑 ∧ 𝑘 ∈ ℕ))
53	49, 52	bitr2i 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) ↔ [𝑘 / 𝑛](𝜑 ∧ 𝑛 ∈ ℕ))
54		sbsbc 3792	. . . . . . . . . . . 12 ⊢ ([𝑘 / 𝑛](𝐹‘𝑛) ⊆ (𝐹‘(𝑛 + 1)) ↔ [𝑘 / 𝑛](𝐹‘𝑛) ⊆ (𝐹‘(𝑛 + 1)))
55		sbcssg 4520	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ V → ([𝑘 / 𝑛](𝐹‘𝑛) ⊆ (𝐹‘(𝑛 + 1)) ↔ ⦋𝑘 / 𝑛⦌(𝐹‘𝑛) ⊆ ⦋𝑘 / 𝑛⦌(𝐹‘(𝑛 + 1))))
56	55	elv 3485	. . . . . . . . . . . 12 ⊢ ([𝑘 / 𝑛](𝐹‘𝑛) ⊆ (𝐹‘(𝑛 + 1)) ↔ ⦋𝑘 / 𝑛⦌(𝐹‘𝑛) ⊆ ⦋𝑘 / 𝑛⦌(𝐹‘(𝑛 + 1)))
57		csbfv 6956	. . . . . . . . . . . . 13 ⊢ ⦋𝑘 / 𝑛⦌(𝐹‘𝑛) = (𝐹‘𝑘)
58		csbfv2g 6955	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ V → ⦋𝑘 / 𝑛⦌(𝐹‘(𝑛 + 1)) = (𝐹‘⦋𝑘 / 𝑛⦌(𝑛 + 1)))
59	58	elv 3485	. . . . . . . . . . . . . 14 ⊢ ⦋𝑘 / 𝑛⦌(𝐹‘(𝑛 + 1)) = (𝐹‘⦋𝑘 / 𝑛⦌(𝑛 + 1))
60		csbov1g 7478	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ V → ⦋𝑘 / 𝑛⦌(𝑛 + 1) = (⦋𝑘 / 𝑛⦌𝑛 + 1))
61	60	elv 3485	. . . . . . . . . . . . . . 15 ⊢ ⦋𝑘 / 𝑛⦌(𝑛 + 1) = (⦋𝑘 / 𝑛⦌𝑛 + 1)
62	61	fveq2i 6909	. . . . . . . . . . . . . 14 ⊢ (𝐹‘⦋𝑘 / 𝑛⦌(𝑛 + 1)) = (𝐹‘(⦋𝑘 / 𝑛⦌𝑛 + 1))
63		vex 3484	. . . . . . . . . . . . . . . . 17 ⊢ 𝑘 ∈ V
64	63	csbvargi 4435	. . . . . . . . . . . . . . . 16 ⊢ ⦋𝑘 / 𝑛⦌𝑛 = 𝑘
65	64	oveq1i 7441	. . . . . . . . . . . . . . 15 ⊢ (⦋𝑘 / 𝑛⦌𝑛 + 1) = (𝑘 + 1)
66	65	fveq2i 6909	. . . . . . . . . . . . . 14 ⊢ (𝐹‘(⦋𝑘 / 𝑛⦌𝑛 + 1)) = (𝐹‘(𝑘 + 1))
67	59, 62, 66	3eqtri 2769	. . . . . . . . . . . . 13 ⊢ ⦋𝑘 / 𝑛⦌(𝐹‘(𝑛 + 1)) = (𝐹‘(𝑘 + 1))
68	57, 67	sseq12i 4014	. . . . . . . . . . . 12 ⊢ (⦋𝑘 / 𝑛⦌(𝐹‘𝑛) ⊆ ⦋𝑘 / 𝑛⦌(𝐹‘(𝑛 + 1)) ↔ (𝐹‘𝑘) ⊆ (𝐹‘(𝑘 + 1)))
69	54, 56, 68	3bitrri 298	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑘) ⊆ (𝐹‘(𝑘 + 1)) ↔ [𝑘 / 𝑛](𝐹‘𝑛) ⊆ (𝐹‘(𝑛 + 1)))
70	53, 69	imbi12i 350	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ⊆ (𝐹‘(𝑘 + 1))) ↔ ([𝑘 / 𝑛](𝜑 ∧ 𝑛 ∈ ℕ) → [𝑘 / 𝑛](𝐹‘𝑛) ⊆ (𝐹‘(𝑛 + 1))))
71	48, 70	bitr4i 278	. . . . . . . . 9 ⊢ ([𝑘 / 𝑛]((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ⊆ (𝐹‘(𝑛 + 1))) ↔ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ⊆ (𝐹‘(𝑘 + 1))))
72	47, 71	mpbi 230	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ⊆ (𝐹‘(𝑘 + 1)))
73		ssequn1 4186	. . . . . . . 8 ⊢ ((𝐹‘𝑘) ⊆ (𝐹‘(𝑘 + 1)) ↔ ((𝐹‘𝑘) ∪ (𝐹‘(𝑘 + 1))) = (𝐹‘(𝑘 + 1)))
74	72, 73	sylib 218	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑘) ∪ (𝐹‘(𝑘 + 1))) = (𝐹‘(𝑘 + 1)))
75	45, 30, 74	syl2anc 584	. . . . . 6 ⊢ (((𝑘 ∈ ℕ ∧ 𝜑) ∧ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) = (𝐹‘𝑘)) → ((𝐹‘𝑘) ∪ (𝐹‘(𝑘 + 1))) = (𝐹‘(𝑘 + 1)))
76	42, 44, 75	3eqtrd 2781	. . . . 5 ⊢ (((𝑘 ∈ ℕ ∧ 𝜑) ∧ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) = (𝐹‘𝑘)) → ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) = (𝐹‘(𝑘 + 1)))
77	76	exp31 419	. . . 4 ⊢ (𝑘 ∈ ℕ → (𝜑 → (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) = (𝐹‘𝑘) → ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) = (𝐹‘(𝑘 + 1)))))
78	77	a2d 29	. . 3 ⊢ (𝑘 ∈ ℕ → ((𝜑 → ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) = (𝐹‘𝑘)) → (𝜑 → ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) = (𝐹‘(𝑘 + 1)))))
79	5, 10, 15, 20, 29, 78	nnind 12284	. 2 ⊢ (𝑖 ∈ ℕ → (𝜑 → ∪ 𝑛 ∈ (1...𝑖)(𝐹‘𝑛) = (𝐹‘𝑖)))
80	79	impcom 407	1 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ∪ 𝑛 ∈ (1...𝑖)(𝐹‘𝑛) = (𝐹‘𝑖))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  [wsb 2064   ∈ wcel 2108  Vcvv 3480  [wsbc 3788  ⦋csb 3899   ∪ cun 3949   ⊆ wss 3951  {csn 4626  ∪ ciun 4991   Fn wfn 6556  ‘cfv 6561  (class class class)co 7431  1c1 11156   + caddc 11158  ℕcn 12266  ℤcz 12613  ℤ_≥cuz 12878  ...cfz 13547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548

This theorem is referenced by:  meascnbl  34220