Theorem iuninc 33197

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 7428	. . . . . 6 ⊢ (𝑗 = 1 → (1...𝑗) = (1...1))
2	1	iuneq1d 4978	. . . . 5 ⊢ (𝑗 = 1 → ∪ 𝑛 ∈ (1...𝑗)(𝐹‘𝑛) = ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛))
3		fveq2 6885	. . . . 5 ⊢ (𝑗 = 1 → (𝐹‘𝑗) = (𝐹‘1))
4	2, 3	eqeq12d 2776	. . . 4 ⊢ (𝑗 = 1 → (∪ 𝑛 ∈ (1...𝑗)(𝐹‘𝑛) = (𝐹‘𝑗) ↔ ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛) = (𝐹‘1)))
5	4	imbi2d 343	. . 3 ⊢ (𝑗 = 1 → ((𝜑 → ∪ 𝑛 ∈ (1...𝑗)(𝐹‘𝑛) = (𝐹‘𝑗)) ↔ (𝜑 → ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛) = (𝐹‘1))))
6		oveq2 7428	. . . . . 6 ⊢ (𝑗 = 𝑘 → (1...𝑗) = (1...𝑘))
7	6	iuneq1d 4978	. . . . 5 ⊢ (𝑗 = 𝑘 → ∪ 𝑛 ∈ (1...𝑗)(𝐹‘𝑛) = ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))
8		fveq2 6885	. . . . 5 ⊢ (𝑗 = 𝑘 → (𝐹‘𝑗) = (𝐹‘𝑘))
9	7, 8	eqeq12d 2776	. . . 4 ⊢ (𝑗 = 𝑘 → (∪ 𝑛 ∈ (1...𝑗)(𝐹‘𝑛) = (𝐹‘𝑗) ↔ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) = (𝐹‘𝑘)))
10	9	imbi2d 343	. . 3 ⊢ (𝑗 = 𝑘 → ((𝜑 → ∪ 𝑛 ∈ (1...𝑗)(𝐹‘𝑛) = (𝐹‘𝑗)) ↔ (𝜑 → ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) = (𝐹‘𝑘))))
11		oveq2 7428	. . . . . 6 ⊢ (𝑗 = (𝑘 + 1) → (1...𝑗) = (1...(𝑘 + 1)))
12	11	iuneq1d 4978	. . . . 5 ⊢ (𝑗 = (𝑘 + 1) → ∪ 𝑛 ∈ (1...𝑗)(𝐹‘𝑛) = ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))
13		fveq2 6885	. . . . 5 ⊢ (𝑗 = (𝑘 + 1) → (𝐹‘𝑗) = (𝐹‘(𝑘 + 1)))
14	12, 13	eqeq12d 2776	. . . 4 ⊢ (𝑗 = (𝑘 + 1) → (∪ 𝑛 ∈ (1...𝑗)(𝐹‘𝑛) = (𝐹‘𝑗) ↔ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) = (𝐹‘(𝑘 + 1))))
15	14	imbi2d 343	. . 3 ⊢ (𝑗 = (𝑘 + 1) → ((𝜑 → ∪ 𝑛 ∈ (1...𝑗)(𝐹‘𝑛) = (𝐹‘𝑗)) ↔ (𝜑 → ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) = (𝐹‘(𝑘 + 1)))))
16		oveq2 7428	. . . . . 6 ⊢ (𝑗 = 𝑖 → (1...𝑗) = (1...𝑖))
17	16	iuneq1d 4978	. . . . 5 ⊢ (𝑗 = 𝑖 → ∪ 𝑛 ∈ (1...𝑗)(𝐹‘𝑛) = ∪ 𝑛 ∈ (1...𝑖)(𝐹‘𝑛))
18		fveq2 6885	. . . . 5 ⊢ (𝑗 = 𝑖 → (𝐹‘𝑗) = (𝐹‘𝑖))
19	17, 18	eqeq12d 2776	. . . 4 ⊢ (𝑗 = 𝑖 → (∪ 𝑛 ∈ (1...𝑗)(𝐹‘𝑛) = (𝐹‘𝑗) ↔ ∪ 𝑛 ∈ (1...𝑖)(𝐹‘𝑛) = (𝐹‘𝑖)))
20	19	imbi2d 343	. . 3 ⊢ (𝑗 = 𝑖 → ((𝜑 → ∪ 𝑛 ∈ (1...𝑗)(𝐹‘𝑛) = (𝐹‘𝑗)) ↔ (𝜑 → ∪ 𝑛 ∈ (1...𝑖)(𝐹‘𝑛) = (𝐹‘𝑖))))
21		1z 12726	. . . . . 6 ⊢ 1 ∈ ℤ
22		fzsn 13700	. . . . . 6 ⊢ (1 ∈ ℤ → (1...1) = {1})
23		iuneq1 4967	. . . . . 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 11303	. . . . . 6 ⊢ 1 ∈ V
26		fveq2 6885	. . . . . 6 ⊢ (𝑛 = 1 → (𝐹‘𝑛) = (𝐹‘1))
27	25, 26	iunxsn 5050	. . . . 5 ⊢ ∪ 𝑛 ∈ {1} (𝐹‘𝑛) = (𝐹‘1)
28	24, 27	eqtri 2783	. . . 4 ⊢ ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛) = (𝐹‘1)
29	28	a1i 11	. . 3 ⊢ (𝜑 → ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛) = (𝐹‘1))
30		simpll 779	. . . . . . 7 ⊢ (((𝑘 ∈ ℕ ∧ 𝜑) ∧ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) = (𝐹‘𝑘)) → 𝑘 ∈ ℕ)
31		elnnuz 13005	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ ↔ 𝑘 ∈ (ℤ≥‘1))
32		fzsuc 13705	. . . . . . . . . 10 ⊢ (𝑘 ∈ (ℤ≥‘1) → (1...(𝑘 + 1)) = ((1...𝑘) ∪ {(𝑘 + 1)}))
33	31, 32	sylbi 220	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (1...(𝑘 + 1)) = ((1...𝑘) ∪ {(𝑘 + 1)}))
34	33	iuneq1d 4978	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) = ∪ 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹‘𝑛))
35		iunxun 5053	. . . . . . . . 9 ⊢ ∪ 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹‘𝑛) = (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ ∪ 𝑛 ∈ {(𝑘 + 1)} (𝐹‘𝑛))
36		ovex 7453	. . . . . . . . . . 11 ⊢ (𝑘 + 1) ∈ V
37		fveq2 6885	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑘 + 1) → (𝐹‘𝑛) = (𝐹‘(𝑘 + 1)))
38	36, 37	iunxsn 5050	. . . . . . . . . 10 ⊢ ∪ 𝑛 ∈ {(𝑘 + 1)} (𝐹‘𝑛) = (𝐹‘(𝑘 + 1))
39	38	uneq2i 4164	. . . . . . . . 9 ⊢ (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ ∪ 𝑛 ∈ {(𝑘 + 1)} (𝐹‘𝑛)) = (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ (𝐹‘(𝑘 + 1)))
40	35, 39	eqtri 2783	. . . . . . . 8 ⊢ ∪ 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹‘𝑛) = (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ (𝐹‘(𝑘 + 1)))
41	34, 40	eqtrdi 2811	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) = (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ (𝐹‘(𝑘 + 1))))
42	30, 41	syl 18	. . . . . 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 490	. . . . . . 7 ⊢ (((𝑘 ∈ ℕ ∧ 𝜑) ∧ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) = (𝐹‘𝑘)) → ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) = (𝐹‘𝑘))
44	43	uneq1d 4166	. . . . . 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 781	. . . . . . 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 2102	. . . . . . . . 9 ⊢ [𝑘 / 𝑛]((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ⊆ (𝐹‘(𝑛 + 1)))
48		sbim 2336	. . . . . . . . . 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 2116	. . . . . . . . . . . 12 ⊢ ([𝑘 / 𝑛](𝜑 ∧ 𝑛 ∈ ℕ) ↔ ([𝑘 / 𝑛]𝜑 ∧ [𝑘 / 𝑛]𝑛 ∈ ℕ))
50		sbv 2124	. . . . . . . . . . . . 13 ⊢ ([𝑘 / 𝑛]𝜑 ↔ 𝜑)
51		clelsb1 2887	. . . . . . . . . . . . 13 ⊢ ([𝑘 / 𝑛]𝑛 ∈ ℕ ↔ 𝑘 ∈ ℕ)
52	50, 51	anbi12i 640	. . . . . . . . . . . 12 ⊢ (([𝑘 / 𝑛]𝜑 ∧ [𝑘 / 𝑛]𝑛 ∈ ℕ) ↔ (𝜑 ∧ 𝑘 ∈ ℕ))
53	49, 52	bitr2i 279	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) ↔ [𝑘 / 𝑛](𝜑 ∧ 𝑛 ∈ ℕ))
54		sbsbc 3742	. . . . . . . . . . . 12 ⊢ ([𝑘 / 𝑛](𝐹‘𝑛) ⊆ (𝐹‘(𝑛 + 1)) ↔ [𝑘 / 𝑛](𝐹‘𝑛) ⊆ (𝐹‘(𝑛 + 1)))
55		sbcssg 4476	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ V → ([𝑘 / 𝑛](𝐹‘𝑛) ⊆ (𝐹‘(𝑛 + 1)) ↔ ⦋𝑘 / 𝑛⦌(𝐹‘𝑛) ⊆ ⦋𝑘 / 𝑛⦌(𝐹‘(𝑛 + 1))))
56	55	elv 3455	. . . . . . . . . . . 12 ⊢ ([𝑘 / 𝑛](𝐹‘𝑛) ⊆ (𝐹‘(𝑛 + 1)) ↔ ⦋𝑘 / 𝑛⦌(𝐹‘𝑛) ⊆ ⦋𝑘 / 𝑛⦌(𝐹‘(𝑛 + 1)))
57		csbfv 6932	. . . . . . . . . . . . 13 ⊢ ⦋𝑘 / 𝑛⦌(𝐹‘𝑛) = (𝐹‘𝑘)
58		csbfv2g 6931	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ V → ⦋𝑘 / 𝑛⦌(𝐹‘(𝑛 + 1)) = (𝐹‘⦋𝑘 / 𝑛⦌(𝑛 + 1)))
59	58	elv 3455	. . . . . . . . . . . . . 14 ⊢ ⦋𝑘 / 𝑛⦌(𝐹‘(𝑛 + 1)) = (𝐹‘⦋𝑘 / 𝑛⦌(𝑛 + 1))
60		csbov1g 7467	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ V → ⦋𝑘 / 𝑛⦌(𝑛 + 1) = (⦋𝑘 / 𝑛⦌𝑛 + 1))
61	60	elv 3455	. . . . . . . . . . . . . . 15 ⊢ ⦋𝑘 / 𝑛⦌(𝑛 + 1) = (⦋𝑘 / 𝑛⦌𝑛 + 1)
62	61	fveq2i 6888	. . . . . . . . . . . . . 14 ⊢ (𝐹‘⦋𝑘 / 𝑛⦌(𝑛 + 1)) = (𝐹‘(⦋𝑘 / 𝑛⦌𝑛 + 1))
63		vex 3454	. . . . . . . . . . . . . . . . 17 ⊢ 𝑘 ∈ V
64	63	csbvargi 4392	. . . . . . . . . . . . . . . 16 ⊢ ⦋𝑘 / 𝑛⦌𝑛 = 𝑘
65	64	oveq1i 7430	. . . . . . . . . . . . . . 15 ⊢ (⦋𝑘 / 𝑛⦌𝑛 + 1) = (𝑘 + 1)
66	65	fveq2i 6888	. . . . . . . . . . . . . 14 ⊢ (𝐹‘(⦋𝑘 / 𝑛⦌𝑛 + 1)) = (𝐹‘(𝑘 + 1))
67	59, 62, 66	3eqtri 2787	. . . . . . . . . . . . 13 ⊢ ⦋𝑘 / 𝑛⦌(𝐹‘(𝑛 + 1)) = (𝐹‘(𝑘 + 1))
68	57, 67	sseq12i 3938	. . . . . . . . . . . 12 ⊢ (⦋𝑘 / 𝑛⦌(𝐹‘𝑛) ⊆ ⦋𝑘 / 𝑛⦌(𝐹‘(𝑛 + 1)) ↔ (𝐹‘𝑘) ⊆ (𝐹‘(𝑘 + 1)))
69	54, 56, 68	3bitrri 301	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑘) ⊆ (𝐹‘(𝑘 + 1)) ↔ [𝑘 / 𝑛](𝐹‘𝑛) ⊆ (𝐹‘(𝑛 + 1)))
70	53, 69	imbi12i 353	. . . . . . . . . 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 281	. . . . . . . . 9 ⊢ ([𝑘 / 𝑛]((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ⊆ (𝐹‘(𝑛 + 1))) ↔ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ⊆ (𝐹‘(𝑘 + 1))))
72	47, 71	mpbi 233	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ⊆ (𝐹‘(𝑘 + 1)))
73		ssequn1 4184	. . . . . . . 8 ⊢ ((𝐹‘𝑘) ⊆ (𝐹‘(𝑘 + 1)) ↔ ((𝐹‘𝑘) ∪ (𝐹‘(𝑘 + 1))) = (𝐹‘(𝑘 + 1)))
74	72, 73	sylib 221	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑘) ∪ (𝐹‘(𝑘 + 1))) = (𝐹‘(𝑘 + 1)))
75	45, 30, 74	syl2anc 596	. . . . . 6 ⊢ (((𝑘 ∈ ℕ ∧ 𝜑) ∧ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) = (𝐹‘𝑘)) → ((𝐹‘𝑘) ∪ (𝐹‘(𝑘 + 1))) = (𝐹‘(𝑘 + 1)))
76	42, 44, 75	3eqtrd 2799	. . . . 5 ⊢ (((𝑘 ∈ ℕ ∧ 𝜑) ∧ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) = (𝐹‘𝑘)) → ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) = (𝐹‘(𝑘 + 1)))
77	76	exp31 425	. . . 4 ⊢ (𝑘 ∈ ℕ → (𝜑 → (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) = (𝐹‘𝑘) → ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) = (𝐹‘(𝑘 + 1)))))
78	77	a2d 30	. . 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 12353	. 2 ⊢ (𝑖 ∈ ℕ → (𝜑 → ∪ 𝑛 ∈ (1...𝑖)(𝐹‘𝑛) = (𝐹‘𝑖)))
80	79	impcom 413	1 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ∪ 𝑛 ∈ (1...𝑖)(𝐹‘𝑛) = (𝐹‘𝑖))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569  [wsb 2098   ∈ wcel 2144  Vcvv 3450  [wsbc 3738  ⦋csb 3846   ⊆ wss 3895   ∪ cun 4149  {csn 4583  ∪ ciun 4950   Fn wfn 6533  ‘cfv 6538  (class class class)co 7420  1c1 11201   + caddc 11203  ℕcn 12335  ℤcz 12693  ℤ_≥cuz 12965  ...cfz 13639

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-n0 12607  df-z 12694  df-uz 12966  df-fz 13640

This theorem is used by:  meascnbl  34894