mbfsup - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > mbfsup		Structured version Visualization version GIF version

Theorem mbfsup 25587

Description: The supremum of a sequence of measurable, real-valued functions is measurable. Note that in this and related theorems, 𝐵(𝑛, 𝑥) is a function of both 𝑛 and 𝑥, since it is an 𝑛-indexed sequence of functions on 𝑥. (Contributed by Mario Carneiro, 14-Aug-2014.) (Revised by Mario Carneiro, 7-Sep-2014.)

**Hypotheses**
Ref	Expression
mbfsup.1	⊢ 𝑍 = (ℤ_≥‘𝑀)
mbfsup.2	⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ))
mbfsup.3	⊢ (𝜑 → 𝑀 ∈ ℤ)
mbfsup.4	⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
mbfsup.5	⊢ ((𝜑 ∧ (𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴)) → 𝐵 ∈ ℝ)
mbfsup.6	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝐵 ≤ 𝑦)

**Assertion**
Ref	Expression
mbfsup	⊢ (𝜑 → 𝐺 ∈ MblFn)

Distinct variable groups: 𝑥,𝑛,𝑦,𝐴 𝑦,𝐵 𝜑,𝑛,𝑥,𝑦 𝑛,𝑍,𝑥,𝑦 Allowed substitution hints: 𝐵(𝑥,𝑛) 𝐺(𝑥,𝑦,𝑛) 𝑀(𝑥,𝑦,𝑛)

Proof of Theorem mbfsup Dummy variables 𝑚 𝑧 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mbfsup.5	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴)) → 𝐵 ∈ ℝ)
2	1	anassrs 467	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
3	2	an32s 652	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → 𝐵 ∈ ℝ)
4	3	fmpttd 7043	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵):𝑍⟶ℝ)
5	4	frnd 6654	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ran (𝑛 ∈ 𝑍 ↦ 𝐵) ⊆ ℝ)
6		mbfsup.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℤ)
7		uzid 12742	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ_≥‘𝑀))
8	6, 7	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘𝑀))
9		mbfsup.1	. . . . . . . . 9 ⊢ 𝑍 = (ℤ_≥‘𝑀)
10	8, 9	eleqtrrdi 2842	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
11	10	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑀 ∈ 𝑍)
12		eqid 2731	. . . . . . . 8 ⊢ (𝑛 ∈ 𝑍 ↦ 𝐵) = (𝑛 ∈ 𝑍 ↦ 𝐵)
13	12, 3	dmmptd 6621	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → dom (𝑛 ∈ 𝑍 ↦ 𝐵) = 𝑍)
14	11, 13	eleqtrrd 2834	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑀 ∈ dom (𝑛 ∈ 𝑍 ↦ 𝐵))
15	14	ne0d 4287	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → dom (𝑛 ∈ 𝑍 ↦ 𝐵) ≠ ∅)
16		dm0rn0 5859	. . . . . 6 ⊢ (dom (𝑛 ∈ 𝑍 ↦ 𝐵) = ∅ ↔ ran (𝑛 ∈ 𝑍 ↦ 𝐵) = ∅)
17	16	necon3bii 2980	. . . . 5 ⊢ (dom (𝑛 ∈ 𝑍 ↦ 𝐵) ≠ ∅ ↔ ran (𝑛 ∈ 𝑍 ↦ 𝐵) ≠ ∅)
18	15, 17	sylib 218	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ran (𝑛 ∈ 𝑍 ↦ 𝐵) ≠ ∅)
19		mbfsup.6	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝐵 ≤ 𝑦)
20	4	ffnd 6647	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵) Fn 𝑍)
21		breq1 5089	. . . . . . . . 9 ⊢ (𝑧 = ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) → (𝑧 ≤ 𝑦 ↔ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) ≤ 𝑦))
22	21	ralrn 7016	. . . . . . . 8 ⊢ ((𝑛 ∈ 𝑍 ↦ 𝐵) Fn 𝑍 → (∀𝑧 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)𝑧 ≤ 𝑦 ↔ ∀𝑚 ∈ 𝑍 ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) ≤ 𝑦))
23	20, 22	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑧 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)𝑧 ≤ 𝑦 ↔ ∀𝑚 ∈ 𝑍 ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) ≤ 𝑦))
24		nffvmpt1 6828	. . . . . . . . . 10 ⊢ Ⅎ𝑛((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚)
25		nfcv 2894	. . . . . . . . . 10 ⊢ Ⅎ𝑛 ≤
26		nfcv 2894	. . . . . . . . . 10 ⊢ Ⅎ𝑛𝑦
27	24, 25, 26	nfbr 5133	. . . . . . . . 9 ⊢ Ⅎ𝑛((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) ≤ 𝑦
28		nfv 1915	. . . . . . . . 9 ⊢ Ⅎ𝑚((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) ≤ 𝑦
29		fveq2 6817	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) = ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛))
30	29	breq1d 5096	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) ≤ 𝑦 ↔ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) ≤ 𝑦))
31	27, 28, 30	cbvralw 3274	. . . . . . . 8 ⊢ (∀𝑚 ∈ 𝑍 ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) ≤ 𝑦 ↔ ∀𝑛 ∈ 𝑍 ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) ≤ 𝑦)
32		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍)
33	12	fvmpt2 6935	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝐵 ∈ ℝ) → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = 𝐵)
34	32, 3, 33	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = 𝐵)
35	34	breq1d 5096	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → (((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) ≤ 𝑦 ↔ 𝐵 ≤ 𝑦))
36	35	ralbidva 3153	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑛 ∈ 𝑍 ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) ≤ 𝑦 ↔ ∀𝑛 ∈ 𝑍 𝐵 ≤ 𝑦))
37	31, 36	bitrid 283	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑚 ∈ 𝑍 ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) ≤ 𝑦 ↔ ∀𝑛 ∈ 𝑍 𝐵 ≤ 𝑦))
38	23, 37	bitrd 279	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑧 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)𝑧 ≤ 𝑦 ↔ ∀𝑛 ∈ 𝑍 𝐵 ≤ 𝑦))
39	38	rexbidv 3156	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)𝑧 ≤ 𝑦 ↔ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝐵 ≤ 𝑦))
40	19, 39	mpbird 257	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)𝑧 ≤ 𝑦)
41	5, 18, 40	suprcld 12080	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → sup(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ) ∈ ℝ)
42		mbfsup.2	. . 3 ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ))
43	41, 42	fmptd 7042	. 2 ⊢ (𝜑 → 𝐺:𝐴⟶ℝ)
44		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
45		ltso 11188	. . . . . . . . . . . . . 14 ⊢ < Or ℝ
46	45	supex 9343	. . . . . . . . . . . . 13 ⊢ sup(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ) ∈ V
47	42	fvmpt2 6935	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ∧ sup(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ) ∈ V) → (𝐺‘𝑥) = sup(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ))
48	44, 46, 47	sylancl 586	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = sup(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ))
49	48	breq2d 5098	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (𝑡 < (𝐺‘𝑥) ↔ 𝑡 < sup(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < )))
50	5, 18, 40	3jca 1128	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ran (𝑛 ∈ 𝑍 ↦ 𝐵) ⊆ ℝ ∧ ran (𝑛 ∈ 𝑍 ↦ 𝐵) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)𝑧 ≤ 𝑦))
51	50	adantlr 715	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (ran (𝑛 ∈ 𝑍 ↦ 𝐵) ⊆ ℝ ∧ ran (𝑛 ∈ 𝑍 ↦ 𝐵) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)𝑧 ≤ 𝑦))
52		simplr 768	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝑡 ∈ ℝ)
53		suprlub 12081	. . . . . . . . . . . 12 ⊢ (((ran (𝑛 ∈ 𝑍 ↦ 𝐵) ⊆ ℝ ∧ ran (𝑛 ∈ 𝑍 ↦ 𝐵) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)𝑧 ≤ 𝑦) ∧ 𝑡 ∈ ℝ) → (𝑡 < sup(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ) ↔ ∃𝑧 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)𝑡 < 𝑧))
54	51, 52, 53	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (𝑡 < sup(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ) ↔ ∃𝑧 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)𝑡 < 𝑧))
55	20	adantlr 715	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵) Fn 𝑍)
56		breq2 5090	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) → (𝑡 < 𝑧 ↔ 𝑡 < ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚)))
57	56	rexrn 7015	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ 𝑍 ↦ 𝐵) Fn 𝑍 → (∃𝑧 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)𝑡 < 𝑧 ↔ ∃𝑚 ∈ 𝑍 𝑡 < ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚)))
58	55, 57	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (∃𝑧 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)𝑡 < 𝑧 ↔ ∃𝑚 ∈ 𝑍 𝑡 < ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚)))
59		nfcv 2894	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛𝑡
60		nfcv 2894	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛 <
61	59, 60, 24	nfbr 5133	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛 𝑡 < ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚)
62		nfv 1915	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑚 𝑡 < ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛)
63	29	breq2d 5098	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → (𝑡 < ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) ↔ 𝑡 < ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛)))
64	61, 62, 63	cbvrexw 3275	. . . . . . . . . . . . 13 ⊢ (∃𝑚 ∈ 𝑍 𝑡 < ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) ↔ ∃𝑛 ∈ 𝑍 𝑡 < ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛))
65	12	fvmpt2i 6934	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ 𝑍 → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = ( I ‘𝐵))
66		eqid 2731	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
67	66	fvmpt2i 6934	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = ( I ‘𝐵))
68	67	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = ( I ‘𝐵))
69	68	eqcomd 2737	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ( I ‘𝐵) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
70	65, 69	sylan9eqr 2788	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
71	70	breq2d 5098	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → (𝑡 < ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) ↔ 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)))
72	71	rexbidva 3154	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∃𝑛 ∈ 𝑍 𝑡 < ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) ↔ ∃𝑛 ∈ 𝑍 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)))
73	72	adantlr 715	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (∃𝑛 ∈ 𝑍 𝑡 < ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) ↔ ∃𝑛 ∈ 𝑍 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)))
74	64, 73	bitrid 283	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (∃𝑚 ∈ 𝑍 𝑡 < ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) ↔ ∃𝑛 ∈ 𝑍 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)))
75	58, 74	bitrd 279	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (∃𝑧 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)𝑡 < 𝑧 ↔ ∃𝑛 ∈ 𝑍 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)))
76	49, 54, 75	3bitrd 305	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (𝑡 < (𝐺‘𝑥) ↔ ∃𝑛 ∈ 𝑍 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)))
77	76	ralrimiva 3124	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → ∀𝑥 ∈ 𝐴 (𝑡 < (𝐺‘𝑥) ↔ ∃𝑛 ∈ 𝑍 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)))
78		nfv 1915	. . . . . . . . . 10 ⊢ Ⅎ𝑧(𝑡 < (𝐺‘𝑥) ↔ ∃𝑛 ∈ 𝑍 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
79		nfcv 2894	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝑡
80		nfcv 2894	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥 <
81		nfmpt1 5185	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ))
82	42, 81	nfcxfr 2892	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥𝐺
83		nfcv 2894	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥𝑧
84	82, 83	nffv 6827	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝐺‘𝑧)
85	79, 80, 84	nfbr 5133	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 𝑡 < (𝐺‘𝑧)
86		nfcv 2894	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝑍
87		nffvmpt1 6828	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧)
88	79, 80, 87	nfbr 5133	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧)
89	86, 88	nfrexw 3280	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∃𝑛 ∈ 𝑍 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧)
90	85, 89	nfbi 1904	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑡 < (𝐺‘𝑧) ↔ ∃𝑛 ∈ 𝑍 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧))
91		fveq2 6817	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝐺‘𝑥) = (𝐺‘𝑧))
92	91	breq2d 5098	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝑡 < (𝐺‘𝑥) ↔ 𝑡 < (𝐺‘𝑧)))
93		fveq2 6817	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧))
94	93	breq2d 5098	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ↔ 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧)))
95	94	rexbidv 3156	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (∃𝑛 ∈ 𝑍 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ↔ ∃𝑛 ∈ 𝑍 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧)))
96	92, 95	bibi12d 345	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → ((𝑡 < (𝐺‘𝑥) ↔ ∃𝑛 ∈ 𝑍 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ↔ (𝑡 < (𝐺‘𝑧) ↔ ∃𝑛 ∈ 𝑍 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧))))
97	78, 90, 96	cbvralw 3274	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 (𝑡 < (𝐺‘𝑥) ↔ ∃𝑛 ∈ 𝑍 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ↔ ∀𝑧 ∈ 𝐴 (𝑡 < (𝐺‘𝑧) ↔ ∃𝑛 ∈ 𝑍 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧)))
98	77, 97	sylib 218	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → ∀𝑧 ∈ 𝐴 (𝑡 < (𝐺‘𝑧) ↔ ∃𝑛 ∈ 𝑍 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧)))
99	98	r19.21bi 3224	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑧 ∈ 𝐴) → (𝑡 < (𝐺‘𝑧) ↔ ∃𝑛 ∈ 𝑍 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧)))
100	43	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → 𝐺:𝐴⟶ℝ)
101	100	ffvelcdmda 7012	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑧 ∈ 𝐴) → (𝐺‘𝑧) ∈ ℝ)
102		rexr 11153	. . . . . . . . . 10 ⊢ (𝑡 ∈ ℝ → 𝑡 ∈ ℝ^*)
103	102	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑧 ∈ 𝐴) → 𝑡 ∈ ℝ^*)
104		elioopnf 13338	. . . . . . . . 9 ⊢ (𝑡 ∈ ℝ^* → ((𝐺‘𝑧) ∈ (𝑡(,)+∞) ↔ ((𝐺‘𝑧) ∈ ℝ ∧ 𝑡 < (𝐺‘𝑧))))
105	103, 104	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑧 ∈ 𝐴) → ((𝐺‘𝑧) ∈ (𝑡(,)+∞) ↔ ((𝐺‘𝑧) ∈ ℝ ∧ 𝑡 < (𝐺‘𝑧))))
106	101, 105	mpbirand 707	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑧 ∈ 𝐴) → ((𝐺‘𝑧) ∈ (𝑡(,)+∞) ↔ 𝑡 < (𝐺‘𝑧)))
107	103	adantr 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑧 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → 𝑡 ∈ ℝ^*)
108		elioopnf 13338	. . . . . . . . . 10 ⊢ (𝑡 ∈ ℝ^* → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ∈ (𝑡(,)+∞) ↔ (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ∈ ℝ ∧ 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧))))
109	107, 108	syl 17	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑧 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ∈ (𝑡(,)+∞) ↔ (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ∈ ℝ ∧ 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧))))
110	2	fmpttd 7043	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ)
111	110	ffvelcdmda 7012	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ∈ ℝ)
112	111	biantrurd 532	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝐴) → (𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ↔ (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ∈ ℝ ∧ 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧))))
113	112	an32s 652	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → (𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ↔ (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ∈ ℝ ∧ 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧))))
114	113	adantllr 719	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑧 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → (𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ↔ (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ∈ ℝ ∧ 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧))))
115	109, 114	bitr4d 282	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑧 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ∈ (𝑡(,)+∞) ↔ 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧)))
116	115	rexbidva 3154	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑧 ∈ 𝐴) → (∃𝑛 ∈ 𝑍 ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ∈ (𝑡(,)+∞) ↔ ∃𝑛 ∈ 𝑍 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧)))
117	99, 106, 116	3bitr4d 311	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑧 ∈ 𝐴) → ((𝐺‘𝑧) ∈ (𝑡(,)+∞) ↔ ∃𝑛 ∈ 𝑍 ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ∈ (𝑡(,)+∞)))
118	117	pm5.32da 579	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → ((𝑧 ∈ 𝐴 ∧ (𝐺‘𝑧) ∈ (𝑡(,)+∞)) ↔ (𝑧 ∈ 𝐴 ∧ ∃𝑛 ∈ 𝑍 ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ∈ (𝑡(,)+∞))))
119	43	ffnd 6647	. . . . . . 7 ⊢ (𝜑 → 𝐺 Fn 𝐴)
120	119	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → 𝐺 Fn 𝐴)
121		elpreima 6986	. . . . . 6 ⊢ (𝐺 Fn 𝐴 → (𝑧 ∈ (^◡𝐺 “ (𝑡(,)+∞)) ↔ (𝑧 ∈ 𝐴 ∧ (𝐺‘𝑧) ∈ (𝑡(,)+∞))))
122	120, 121	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → (𝑧 ∈ (^◡𝐺 “ (𝑡(,)+∞)) ↔ (𝑧 ∈ 𝐴 ∧ (𝐺‘𝑧) ∈ (𝑡(,)+∞))))
123		eliun 4940	. . . . . 6 ⊢ (𝑧 ∈ ∪ 𝑛 ∈ 𝑍 (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑡(,)+∞)) ↔ ∃𝑛 ∈ 𝑍 𝑧 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑡(,)+∞)))
124	110	ffnd 6647	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
125		elpreima 6986	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 → (𝑧 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑡(,)+∞)) ↔ (𝑧 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ∈ (𝑡(,)+∞))))
126	124, 125	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑧 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑡(,)+∞)) ↔ (𝑧 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ∈ (𝑡(,)+∞))))
127	126	rexbidva 3154	. . . . . . . 8 ⊢ (𝜑 → (∃𝑛 ∈ 𝑍 𝑧 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑡(,)+∞)) ↔ ∃𝑛 ∈ 𝑍 (𝑧 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ∈ (𝑡(,)+∞))))
128	127	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → (∃𝑛 ∈ 𝑍 𝑧 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑡(,)+∞)) ↔ ∃𝑛 ∈ 𝑍 (𝑧 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ∈ (𝑡(,)+∞))))
129		r19.42v 3164	. . . . . . 7 ⊢ (∃𝑛 ∈ 𝑍 (𝑧 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ∈ (𝑡(,)+∞)) ↔ (𝑧 ∈ 𝐴 ∧ ∃𝑛 ∈ 𝑍 ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ∈ (𝑡(,)+∞)))
130	128, 129	bitrdi 287	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → (∃𝑛 ∈ 𝑍 𝑧 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑡(,)+∞)) ↔ (𝑧 ∈ 𝐴 ∧ ∃𝑛 ∈ 𝑍 ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ∈ (𝑡(,)+∞))))
131	123, 130	bitrid 283	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → (𝑧 ∈ ∪ 𝑛 ∈ 𝑍 (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑡(,)+∞)) ↔ (𝑧 ∈ 𝐴 ∧ ∃𝑛 ∈ 𝑍 ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ∈ (𝑡(,)+∞))))
132	118, 122, 131	3bitr4d 311	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → (𝑧 ∈ (^◡𝐺 “ (𝑡(,)+∞)) ↔ 𝑧 ∈ ∪ 𝑛 ∈ 𝑍 (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑡(,)+∞))))
133	132	eqrdv 2729	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → (^◡𝐺 “ (𝑡(,)+∞)) = ∪ 𝑛 ∈ 𝑍 (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑡(,)+∞)))
134		zex 12472	. . . . . . 7 ⊢ ℤ ∈ V
135		uzssz 12748	. . . . . . 7 ⊢ (ℤ_≥‘𝑀) ⊆ ℤ
136		ssdomg 8917	. . . . . . 7 ⊢ (ℤ ∈ V → ((ℤ_≥‘𝑀) ⊆ ℤ → (ℤ_≥‘𝑀) ≼ ℤ))
137	134, 135, 136	mp2 9	. . . . . 6 ⊢ (ℤ_≥‘𝑀) ≼ ℤ
138	9, 137	eqbrtri 5107	. . . . 5 ⊢ 𝑍 ≼ ℤ
139		znnen 16116	. . . . 5 ⊢ ℤ ≈ ℕ
140		domentr 8930	. . . . 5 ⊢ ((𝑍 ≼ ℤ ∧ ℤ ≈ ℕ) → 𝑍 ≼ ℕ)
141	138, 139, 140	mp2an 692	. . . 4 ⊢ 𝑍 ≼ ℕ
142		mbfsup.4	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
143		mbfima 25553	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ) → (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑡(,)+∞)) ∈ dom vol)
144	142, 110, 143	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑡(,)+∞)) ∈ dom vol)
145	144	ralrimiva 3124	. . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ 𝑍 (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑡(,)+∞)) ∈ dom vol)
146	145	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → ∀𝑛 ∈ 𝑍 (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑡(,)+∞)) ∈ dom vol)
147		iunmbl2 25480	. . . 4 ⊢ ((𝑍 ≼ ℕ ∧ ∀𝑛 ∈ 𝑍 (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑡(,)+∞)) ∈ dom vol) → ∪ 𝑛 ∈ 𝑍 (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑡(,)+∞)) ∈ dom vol)
148	141, 146, 147	sylancr 587	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → ∪ 𝑛 ∈ 𝑍 (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑡(,)+∞)) ∈ dom vol)
149	133, 148	eqeltrd 2831	. 2 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → (^◡𝐺 “ (𝑡(,)+∞)) ∈ dom vol)
150	43, 149	ismbf3d 25577	1 ⊢ (𝜑 → 𝐺 ∈ MblFn)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∀wral 3047 ∃wrex 3056 Vcvv 3436 ⊆ wss 3897 ∅c0 4278 ∪ ciun 4936 class class class wbr 5086 ↦ cmpt 5167 I cid 5505 ^◡ccnv 5610 dom cdm 5611 ran crn 5612 “ cima 5614 Fn wfn 6471 ⟶wf 6472 ‘cfv 6476 (class class class)co 7341 ≈ cen 8861 ≼ cdom 8862 supcsup 9319 ℝcr 11000 +∞cpnf 11138 ℝ^*cxr 11140 < clt 11141 ≤ cle 11142 ℕcn 12120 ℤcz 12463 ℤ_≥cuz 12727 (,)cioo 13240 volcvol 25386 MblFncmbf 25537

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5212 ax-sep 5229 ax-nul 5239 ax-pow 5298 ax-pr 5365 ax-un 7663 ax-inf2 9526 ax-cc 10321 ax-cnex 11057 ax-resscn 11058 ax-1cn 11059 ax-icn 11060 ax-addcl 11061 ax-addrcl 11062 ax-mulcl 11063 ax-mulrcl 11064 ax-mulcom 11065 ax-addass 11066 ax-mulass 11067 ax-distr 11068 ax-i2m1 11069 ax-1ne0 11070 ax-1rid 11071 ax-rnegex 11072 ax-rrecex 11073 ax-cnre 11074 ax-pre-lttri 11075 ax-pre-lttrn 11076 ax-pre-ltadd 11077 ax-pre-mulgt0 11078 ax-pre-sup 11079

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4279 df-if 4471 df-pw 4547 df-sn 4572 df-pr 4574 df-op 4578 df-uni 4855 df-int 4893 df-iun 4938 df-disj 5054 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5506 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5564 df-se 5565 df-we 5566 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-isom 6485 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-of 7605 df-om 7792 df-1st 7916 df-2nd 7917 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-2o 8381 df-oadd 8384 df-omul 8385 df-er 8617 df-map 8747 df-pm 8748 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-sup 9321 df-inf 9322 df-oi 9391 df-dju 9789 df-card 9827 df-acn 9830 df-pnf 11143 df-mnf 11144 df-xr 11145 df-ltxr 11146 df-le 11147 df-sub 11341 df-neg 11342 df-div 11770 df-nn 12121 df-2 12183 df-3 12184 df-n0 12377 df-z 12464 df-uz 12728 df-q 12842 df-rp 12886 df-xadd 13007 df-ioo 13244 df-ioc 13245 df-ico 13246 df-icc 13247 df-fz 13403 df-fzo 13550 df-fl 13691 df-seq 13904 df-exp 13964 df-hash 14233 df-cj 15001 df-re 15002 df-im 15003 df-sqrt 15137 df-abs 15138 df-clim 15390 df-rlim 15391 df-sum 15589 df-xmet 21279 df-met 21280 df-ovol 25387 df-vol 25388 df-mbf 25542

This theorem is referenced by: mbfinf 25588 mbflimsup 25589

W3C validator