Theorem iuninc 32546

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 7418	. . . . . 6 ⊢ (𝑗 = 1 → (1...𝑗) = (1...1))
2	1	iuneq1d 5000	. . . . 5 ⊢ (𝑗 = 1 → ∪ 𝑛 ∈ (1...𝑗)(𝐹‘𝑛) = ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛))
3		fveq2 6881	. . . . 5 ⊢ (𝑗 = 1 → (𝐹‘𝑗) = (𝐹‘1))
4	2, 3	eqeq12d 2752	. . . 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 7418	. . . . . 6 ⊢ (𝑗 = 𝑘 → (1...𝑗) = (1...𝑘))
7	6	iuneq1d 5000	. . . . 5 ⊢ (𝑗 = 𝑘 → ∪ 𝑛 ∈ (1...𝑗)(𝐹‘𝑛) = ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))
8		fveq2 6881	. . . . 5 ⊢ (𝑗 = 𝑘 → (𝐹‘𝑗) = (𝐹‘𝑘))
9	7, 8	eqeq12d 2752	. . . 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 7418	. . . . . 6 ⊢ (𝑗 = (𝑘 + 1) → (1...𝑗) = (1...(𝑘 + 1)))
12	11	iuneq1d 5000	. . . . 5 ⊢ (𝑗 = (𝑘 + 1) → ∪ 𝑛 ∈ (1...𝑗)(𝐹‘𝑛) = ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))
13		fveq2 6881	. . . . 5 ⊢ (𝑗 = (𝑘 + 1) → (𝐹‘𝑗) = (𝐹‘(𝑘 + 1)))
14	12, 13	eqeq12d 2752	. . . 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 7418	. . . . . 6 ⊢ (𝑗 = 𝑖 → (1...𝑗) = (1...𝑖))
17	16	iuneq1d 5000	. . . . 5 ⊢ (𝑗 = 𝑖 → ∪ 𝑛 ∈ (1...𝑗)(𝐹‘𝑛) = ∪ 𝑛 ∈ (1...𝑖)(𝐹‘𝑛))
18		fveq2 6881	. . . . 5 ⊢ (𝑗 = 𝑖 → (𝐹‘𝑗) = (𝐹‘𝑖))
19	17, 18	eqeq12d 2752	. . . 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 12627	. . . . . 6 ⊢ 1 ∈ ℤ
22		fzsn 13588	. . . . . 6 ⊢ (1 ∈ ℤ → (1...1) = {1})
23		iuneq1 4989	. . . . . 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 11236	. . . . . 6 ⊢ 1 ∈ V
26		fveq2 6881	. . . . . 6 ⊢ (𝑛 = 1 → (𝐹‘𝑛) = (𝐹‘1))
27	25, 26	iunxsn 5072	. . . . 5 ⊢ ∪ 𝑛 ∈ {1} (𝐹‘𝑛) = (𝐹‘1)
28	24, 27	eqtri 2759	. . . 4 ⊢ ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛) = (𝐹‘1)
29	28	a1i 11	. . 3 ⊢ (𝜑 → ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛) = (𝐹‘1))
30		simpll 766	. . . . . . 7 ⊢ (((𝑘 ∈ ℕ ∧ 𝜑) ∧ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) = (𝐹‘𝑘)) → 𝑘 ∈ ℕ)
31		elnnuz 12901	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ ↔ 𝑘 ∈ (ℤ≥‘1))
32		fzsuc 13593	. . . . . . . . . 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 5000	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) = ∪ 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹‘𝑛))
35		iunxun 5075	. . . . . . . . 9 ⊢ ∪ 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹‘𝑛) = (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ ∪ 𝑛 ∈ {(𝑘 + 1)} (𝐹‘𝑛))
36		ovex 7443	. . . . . . . . . . 11 ⊢ (𝑘 + 1) ∈ V
37		fveq2 6881	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑘 + 1) → (𝐹‘𝑛) = (𝐹‘(𝑘 + 1)))
38	36, 37	iunxsn 5072	. . . . . . . . . 10 ⊢ ∪ 𝑛 ∈ {(𝑘 + 1)} (𝐹‘𝑛) = (𝐹‘(𝑘 + 1))
39	38	uneq2i 4145	. . . . . . . . 9 ⊢ (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ ∪ 𝑛 ∈ {(𝑘 + 1)} (𝐹‘𝑛)) = (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ (𝐹‘(𝑘 + 1)))
40	35, 39	eqtri 2759	. . . . . . . 8 ⊢ ∪ 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹‘𝑛) = (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ (𝐹‘(𝑘 + 1)))
41	34, 40	eqtrdi 2787	. . . . . . 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 4147	. . . . . 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 768	. . . . . . 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 2067	. . . . . . . . 9 ⊢ [𝑘 / 𝑛]((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ⊆ (𝐹‘(𝑛 + 1)))
48		sbim 2304	. . . . . . . . . 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 2081	. . . . . . . . . . . 12 ⊢ ([𝑘 / 𝑛](𝜑 ∧ 𝑛 ∈ ℕ) ↔ ([𝑘 / 𝑛]𝜑 ∧ [𝑘 / 𝑛]𝑛 ∈ ℕ))
50		sbv 2089	. . . . . . . . . . . . 13 ⊢ ([𝑘 / 𝑛]𝜑 ↔ 𝜑)
51		clelsb1 2862	. . . . . . . . . . . . 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 3774	. . . . . . . . . . . 12 ⊢ ([𝑘 / 𝑛](𝐹‘𝑛) ⊆ (𝐹‘(𝑛 + 1)) ↔ [𝑘 / 𝑛](𝐹‘𝑛) ⊆ (𝐹‘(𝑛 + 1)))
55		sbcssg 4500	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ V → ([𝑘 / 𝑛](𝐹‘𝑛) ⊆ (𝐹‘(𝑛 + 1)) ↔ ⦋𝑘 / 𝑛⦌(𝐹‘𝑛) ⊆ ⦋𝑘 / 𝑛⦌(𝐹‘(𝑛 + 1))))
56	55	elv 3469	. . . . . . . . . . . 12 ⊢ ([𝑘 / 𝑛](𝐹‘𝑛) ⊆ (𝐹‘(𝑛 + 1)) ↔ ⦋𝑘 / 𝑛⦌(𝐹‘𝑛) ⊆ ⦋𝑘 / 𝑛⦌(𝐹‘(𝑛 + 1)))
57		csbfv 6931	. . . . . . . . . . . . 13 ⊢ ⦋𝑘 / 𝑛⦌(𝐹‘𝑛) = (𝐹‘𝑘)
58		csbfv2g 6930	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ V → ⦋𝑘 / 𝑛⦌(𝐹‘(𝑛 + 1)) = (𝐹‘⦋𝑘 / 𝑛⦌(𝑛 + 1)))
59	58	elv 3469	. . . . . . . . . . . . . 14 ⊢ ⦋𝑘 / 𝑛⦌(𝐹‘(𝑛 + 1)) = (𝐹‘⦋𝑘 / 𝑛⦌(𝑛 + 1))
60		csbov1g 7457	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ V → ⦋𝑘 / 𝑛⦌(𝑛 + 1) = (⦋𝑘 / 𝑛⦌𝑛 + 1))
61	60	elv 3469	. . . . . . . . . . . . . . 15 ⊢ ⦋𝑘 / 𝑛⦌(𝑛 + 1) = (⦋𝑘 / 𝑛⦌𝑛 + 1)
62	61	fveq2i 6884	. . . . . . . . . . . . . 14 ⊢ (𝐹‘⦋𝑘 / 𝑛⦌(𝑛 + 1)) = (𝐹‘(⦋𝑘 / 𝑛⦌𝑛 + 1))
63		vex 3468	. . . . . . . . . . . . . . . . 17 ⊢ 𝑘 ∈ V
64	63	csbvargi 4415	. . . . . . . . . . . . . . . 16 ⊢ ⦋𝑘 / 𝑛⦌𝑛 = 𝑘
65	64	oveq1i 7420	. . . . . . . . . . . . . . 15 ⊢ (⦋𝑘 / 𝑛⦌𝑛 + 1) = (𝑘 + 1)
66	65	fveq2i 6884	. . . . . . . . . . . . . 14 ⊢ (𝐹‘(⦋𝑘 / 𝑛⦌𝑛 + 1)) = (𝐹‘(𝑘 + 1))
67	59, 62, 66	3eqtri 2763	. . . . . . . . . . . . 13 ⊢ ⦋𝑘 / 𝑛⦌(𝐹‘(𝑛 + 1)) = (𝐹‘(𝑘 + 1))
68	57, 67	sseq12i 3994	. . . . . . . . . . . 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 4166	. . . . . . . 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 2775	. . . . 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 12263	. 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 2065   ∈ wcel 2109  Vcvv 3464  [wsbc 3770  ⦋csb 3879   ∪ cun 3929   ⊆ wss 3931  {csn 4606  ∪ ciun 4972   Fn wfn 6531  ‘cfv 6536  (class class class)co 7410  1c1 11135   + caddc 11137  ℕcn 12245  ℤcz 12593  ℤ_≥cuz 12857  ...cfz 13529

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-nn 12246  df-n0 12507  df-z 12594  df-uz 12858  df-fz 13530

This theorem is referenced by:  meascnbl  34255