Theorem iuninc 30099

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 6983	. . . . . 6 ⊢ (𝑗 = 1 → (1...𝑗) = (1...1))
2	1	iuneq1d 4815	. . . . 5 ⊢ (𝑗 = 1 → ∪ 𝑛 ∈ (1...𝑗)(𝐹‘𝑛) = ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛))
3		fveq2 6497	. . . . 5 ⊢ (𝑗 = 1 → (𝐹‘𝑗) = (𝐹‘1))
4	2, 3	eqeq12d 2788	. . . 4 ⊢ (𝑗 = 1 → (∪ 𝑛 ∈ (1...𝑗)(𝐹‘𝑛) = (𝐹‘𝑗) ↔ ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛) = (𝐹‘1)))
5	4	imbi2d 333	. . 3 ⊢ (𝑗 = 1 → ((𝜑 → ∪ 𝑛 ∈ (1...𝑗)(𝐹‘𝑛) = (𝐹‘𝑗)) ↔ (𝜑 → ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛) = (𝐹‘1))))
6		oveq2 6983	. . . . . 6 ⊢ (𝑗 = 𝑘 → (1...𝑗) = (1...𝑘))
7	6	iuneq1d 4815	. . . . 5 ⊢ (𝑗 = 𝑘 → ∪ 𝑛 ∈ (1...𝑗)(𝐹‘𝑛) = ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))
8		fveq2 6497	. . . . 5 ⊢ (𝑗 = 𝑘 → (𝐹‘𝑗) = (𝐹‘𝑘))
9	7, 8	eqeq12d 2788	. . . 4 ⊢ (𝑗 = 𝑘 → (∪ 𝑛 ∈ (1...𝑗)(𝐹‘𝑛) = (𝐹‘𝑗) ↔ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) = (𝐹‘𝑘)))
10	9	imbi2d 333	. . 3 ⊢ (𝑗 = 𝑘 → ((𝜑 → ∪ 𝑛 ∈ (1...𝑗)(𝐹‘𝑛) = (𝐹‘𝑗)) ↔ (𝜑 → ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) = (𝐹‘𝑘))))
11		oveq2 6983	. . . . . 6 ⊢ (𝑗 = (𝑘 + 1) → (1...𝑗) = (1...(𝑘 + 1)))
12	11	iuneq1d 4815	. . . . 5 ⊢ (𝑗 = (𝑘 + 1) → ∪ 𝑛 ∈ (1...𝑗)(𝐹‘𝑛) = ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))
13		fveq2 6497	. . . . 5 ⊢ (𝑗 = (𝑘 + 1) → (𝐹‘𝑗) = (𝐹‘(𝑘 + 1)))
14	12, 13	eqeq12d 2788	. . . 4 ⊢ (𝑗 = (𝑘 + 1) → (∪ 𝑛 ∈ (1...𝑗)(𝐹‘𝑛) = (𝐹‘𝑗) ↔ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) = (𝐹‘(𝑘 + 1))))
15	14	imbi2d 333	. . 3 ⊢ (𝑗 = (𝑘 + 1) → ((𝜑 → ∪ 𝑛 ∈ (1...𝑗)(𝐹‘𝑛) = (𝐹‘𝑗)) ↔ (𝜑 → ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) = (𝐹‘(𝑘 + 1)))))
16		oveq2 6983	. . . . . 6 ⊢ (𝑗 = 𝑖 → (1...𝑗) = (1...𝑖))
17	16	iuneq1d 4815	. . . . 5 ⊢ (𝑗 = 𝑖 → ∪ 𝑛 ∈ (1...𝑗)(𝐹‘𝑛) = ∪ 𝑛 ∈ (1...𝑖)(𝐹‘𝑛))
18		fveq2 6497	. . . . 5 ⊢ (𝑗 = 𝑖 → (𝐹‘𝑗) = (𝐹‘𝑖))
19	17, 18	eqeq12d 2788	. . . 4 ⊢ (𝑗 = 𝑖 → (∪ 𝑛 ∈ (1...𝑗)(𝐹‘𝑛) = (𝐹‘𝑗) ↔ ∪ 𝑛 ∈ (1...𝑖)(𝐹‘𝑛) = (𝐹‘𝑖)))
20	19	imbi2d 333	. . 3 ⊢ (𝑗 = 𝑖 → ((𝜑 → ∪ 𝑛 ∈ (1...𝑗)(𝐹‘𝑛) = (𝐹‘𝑗)) ↔ (𝜑 → ∪ 𝑛 ∈ (1...𝑖)(𝐹‘𝑛) = (𝐹‘𝑖))))
21		1z 11824	. . . . . 6 ⊢ 1 ∈ ℤ
22		fzsn 12764	. . . . . 6 ⊢ (1 ∈ ℤ → (1...1) = {1})
23		iuneq1 4804	. . . . . 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 10434	. . . . . 6 ⊢ 1 ∈ V
26		fveq2 6497	. . . . . 6 ⊢ (𝑛 = 1 → (𝐹‘𝑛) = (𝐹‘1))
27	25, 26	iunxsn 4876	. . . . 5 ⊢ ∪ 𝑛 ∈ {1} (𝐹‘𝑛) = (𝐹‘1)
28	24, 27	eqtri 2797	. . . 4 ⊢ ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛) = (𝐹‘1)
29	28	a1i 11	. . 3 ⊢ (𝜑 → ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛) = (𝐹‘1))
30		simpll 755	. . . . . . 7 ⊢ (((𝑘 ∈ ℕ ∧ 𝜑) ∧ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) = (𝐹‘𝑘)) → 𝑘 ∈ ℕ)
31		elnnuz 12095	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ ↔ 𝑘 ∈ (ℤ≥‘1))
32		fzsuc 12769	. . . . . . . . . 10 ⊢ (𝑘 ∈ (ℤ≥‘1) → (1...(𝑘 + 1)) = ((1...𝑘) ∪ {(𝑘 + 1)}))
33	31, 32	sylbi 209	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (1...(𝑘 + 1)) = ((1...𝑘) ∪ {(𝑘 + 1)}))
34	33	iuneq1d 4815	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) = ∪ 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹‘𝑛))
35		iunxun 4879	. . . . . . . . 9 ⊢ ∪ 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹‘𝑛) = (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ ∪ 𝑛 ∈ {(𝑘 + 1)} (𝐹‘𝑛))
36		ovex 7007	. . . . . . . . . . 11 ⊢ (𝑘 + 1) ∈ V
37		fveq2 6497	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑘 + 1) → (𝐹‘𝑛) = (𝐹‘(𝑘 + 1)))
38	36, 37	iunxsn 4876	. . . . . . . . . 10 ⊢ ∪ 𝑛 ∈ {(𝑘 + 1)} (𝐹‘𝑛) = (𝐹‘(𝑘 + 1))
39	38	uneq2i 4020	. . . . . . . . 9 ⊢ (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ ∪ 𝑛 ∈ {(𝑘 + 1)} (𝐹‘𝑛)) = (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ (𝐹‘(𝑘 + 1)))
40	35, 39	eqtri 2797	. . . . . . . 8 ⊢ ∪ 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹‘𝑛) = (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ (𝐹‘(𝑘 + 1)))
41	34, 40	syl6eq 2825	. . . . . . 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 477	. . . . . . 7 ⊢ (((𝑘 ∈ ℕ ∧ 𝜑) ∧ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) = (𝐹‘𝑘)) → ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) = (𝐹‘𝑘))
44	43	uneq1d 4022	. . . . . 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 757	. . . . . . 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 2018	. . . . . . . . 9 ⊢ [𝑘 / 𝑛]((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ⊆ (𝐹‘(𝑛 + 1)))
48		sbim 2238	. . . . . . . . . 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 2032	. . . . . . . . . . . 12 ⊢ ([𝑘 / 𝑛](𝜑 ∧ 𝑛 ∈ ℕ) ↔ ([𝑘 / 𝑛]𝜑 ∧ [𝑘 / 𝑛]𝑛 ∈ ℕ))
50		sbv 2041	. . . . . . . . . . . . 13 ⊢ ([𝑘 / 𝑛]𝜑 ↔ 𝜑)
51		clelsb3 2888	. . . . . . . . . . . . 13 ⊢ ([𝑘 / 𝑛]𝑛 ∈ ℕ ↔ 𝑘 ∈ ℕ)
52	50, 51	anbi12i 618	. . . . . . . . . . . 12 ⊢ (([𝑘 / 𝑛]𝜑 ∧ [𝑘 / 𝑛]𝑛 ∈ ℕ) ↔ (𝜑 ∧ 𝑘 ∈ ℕ))
53	49, 52	bitr2i 268	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) ↔ [𝑘 / 𝑛](𝜑 ∧ 𝑛 ∈ ℕ))
54		sbsbc 3680	. . . . . . . . . . . 12 ⊢ ([𝑘 / 𝑛](𝐹‘𝑛) ⊆ (𝐹‘(𝑛 + 1)) ↔ [𝑘 / 𝑛](𝐹‘𝑛) ⊆ (𝐹‘(𝑛 + 1)))
55		sbcssg 4341	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ V → ([𝑘 / 𝑛](𝐹‘𝑛) ⊆ (𝐹‘(𝑛 + 1)) ↔ ⦋𝑘 / 𝑛⦌(𝐹‘𝑛) ⊆ ⦋𝑘 / 𝑛⦌(𝐹‘(𝑛 + 1))))
56	55	elv 3415	. . . . . . . . . . . 12 ⊢ ([𝑘 / 𝑛](𝐹‘𝑛) ⊆ (𝐹‘(𝑛 + 1)) ↔ ⦋𝑘 / 𝑛⦌(𝐹‘𝑛) ⊆ ⦋𝑘 / 𝑛⦌(𝐹‘(𝑛 + 1)))
57		csbfv 6543	. . . . . . . . . . . . 13 ⊢ ⦋𝑘 / 𝑛⦌(𝐹‘𝑛) = (𝐹‘𝑘)
58		csbfv2g 6542	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ V → ⦋𝑘 / 𝑛⦌(𝐹‘(𝑛 + 1)) = (𝐹‘⦋𝑘 / 𝑛⦌(𝑛 + 1)))
59	58	elv 3415	. . . . . . . . . . . . . 14 ⊢ ⦋𝑘 / 𝑛⦌(𝐹‘(𝑛 + 1)) = (𝐹‘⦋𝑘 / 𝑛⦌(𝑛 + 1))
60		csbov1g 7019	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ V → ⦋𝑘 / 𝑛⦌(𝑛 + 1) = (⦋𝑘 / 𝑛⦌𝑛 + 1))
61	60	elv 3415	. . . . . . . . . . . . . . 15 ⊢ ⦋𝑘 / 𝑛⦌(𝑛 + 1) = (⦋𝑘 / 𝑛⦌𝑛 + 1)
62	61	fveq2i 6500	. . . . . . . . . . . . . 14 ⊢ (𝐹‘⦋𝑘 / 𝑛⦌(𝑛 + 1)) = (𝐹‘(⦋𝑘 / 𝑛⦌𝑛 + 1))
63		vex 3413	. . . . . . . . . . . . . . . . 17 ⊢ 𝑘 ∈ V
64	63	csbvargi 4263	. . . . . . . . . . . . . . . 16 ⊢ ⦋𝑘 / 𝑛⦌𝑛 = 𝑘
65	64	oveq1i 6985	. . . . . . . . . . . . . . 15 ⊢ (⦋𝑘 / 𝑛⦌𝑛 + 1) = (𝑘 + 1)
66	65	fveq2i 6500	. . . . . . . . . . . . . 14 ⊢ (𝐹‘(⦋𝑘 / 𝑛⦌𝑛 + 1)) = (𝐹‘(𝑘 + 1))
67	59, 62, 66	3eqtri 2801	. . . . . . . . . . . . 13 ⊢ ⦋𝑘 / 𝑛⦌(𝐹‘(𝑛 + 1)) = (𝐹‘(𝑘 + 1))
68	57, 67	sseq12i 3882	. . . . . . . . . . . 12 ⊢ (⦋𝑘 / 𝑛⦌(𝐹‘𝑛) ⊆ ⦋𝑘 / 𝑛⦌(𝐹‘(𝑛 + 1)) ↔ (𝐹‘𝑘) ⊆ (𝐹‘(𝑘 + 1)))
69	54, 56, 68	3bitrri 290	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑘) ⊆ (𝐹‘(𝑘 + 1)) ↔ [𝑘 / 𝑛](𝐹‘𝑛) ⊆ (𝐹‘(𝑛 + 1)))
70	53, 69	imbi12i 343	. . . . . . . . . 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 270	. . . . . . . . 9 ⊢ ([𝑘 / 𝑛]((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ⊆ (𝐹‘(𝑛 + 1))) ↔ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ⊆ (𝐹‘(𝑘 + 1))))
72	47, 71	mpbi 222	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ⊆ (𝐹‘(𝑘 + 1)))
73		ssequn1 4039	. . . . . . . 8 ⊢ ((𝐹‘𝑘) ⊆ (𝐹‘(𝑘 + 1)) ↔ ((𝐹‘𝑘) ∪ (𝐹‘(𝑘 + 1))) = (𝐹‘(𝑘 + 1)))
74	72, 73	sylib 210	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑘) ∪ (𝐹‘(𝑘 + 1))) = (𝐹‘(𝑘 + 1)))
75	45, 30, 74	syl2anc 576	. . . . . 6 ⊢ (((𝑘 ∈ ℕ ∧ 𝜑) ∧ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) = (𝐹‘𝑘)) → ((𝐹‘𝑘) ∪ (𝐹‘(𝑘 + 1))) = (𝐹‘(𝑘 + 1)))
76	42, 44, 75	3eqtrd 2813	. . . . 5 ⊢ (((𝑘 ∈ ℕ ∧ 𝜑) ∧ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) = (𝐹‘𝑘)) → ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) = (𝐹‘(𝑘 + 1)))
77	76	exp31 412	. . . 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 11458	. 2 ⊢ (𝑖 ∈ ℕ → (𝜑 → ∪ 𝑛 ∈ (1...𝑖)(𝐹‘𝑛) = (𝐹‘𝑖)))
80	79	impcom 399	1 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ∪ 𝑛 ∈ (1...𝑖)(𝐹‘𝑛) = (𝐹‘𝑖))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   = wceq 1508  [wsb 2016   ∈ wcel 2051  Vcvv 3410  [wsbc 3676  ⦋csb 3781   ∪ cun 3822   ⊆ wss 3824  {csn 4436  ∪ ciun 4789   Fn wfn 6181  ‘cfv 6186  (class class class)co 6975  1c1 10335   + caddc 10337  ℕcn 11438  ℤcz 11792  ℤ_≥cuz 12057  ...cfz 12707

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-sep 5057  ax-nul 5064  ax-pow 5116  ax-pr 5183  ax-un 7278  ax-cnex 10390  ax-resscn 10391  ax-1cn 10392  ax-icn 10393  ax-addcl 10394  ax-addrcl 10395  ax-mulcl 10396  ax-mulrcl 10397  ax-mulcom 10398  ax-addass 10399  ax-mulass 10400  ax-distr 10401  ax-i2m1 10402  ax-1ne0 10403  ax-1rid 10404  ax-rnegex 10405  ax-rrecex 10406  ax-cnre 10407  ax-pre-lttri 10408  ax-pre-lttrn 10409  ax-pre-ltadd 10410  ax-pre-mulgt0 10411

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-fal 1521  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ne 2963  df-nel 3069  df-ral 3088  df-rex 3089  df-reu 3090  df-rab 3092  df-v 3412  df-sbc 3677  df-csb 3782  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-pss 3840  df-nul 4174  df-if 4346  df-pw 4419  df-sn 4437  df-pr 4439  df-tp 4441  df-op 4443  df-uni 4710  df-iun 4791  df-br 4927  df-opab 4989  df-mpt 5006  df-tr 5028  df-id 5309  df-eprel 5314  df-po 5323  df-so 5324  df-fr 5363  df-we 5365  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-pred 5984  df-ord 6030  df-on 6031  df-lim 6032  df-suc 6033  df-iota 6150  df-fun 6188  df-fn 6189  df-f 6190  df-f1 6191  df-fo 6192  df-f1o 6193  df-fv 6194  df-riota 6936  df-ov 6978  df-oprab 6979  df-mpo 6980  df-om 7396  df-1st 7500  df-2nd 7501  df-wrecs 7749  df-recs 7811  df-rdg 7849  df-er 8088  df-en 8306  df-dom 8307  df-sdom 8308  df-pnf 10475  df-mnf 10476  df-xr 10477  df-ltxr 10478  df-le 10479  df-sub 10671  df-neg 10672  df-nn 11439  df-n0 11707  df-z 11793  df-uz 12058  df-fz 12708

This theorem is referenced by:  meascnbl  31156