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Theorem mbfsup 24259

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 470	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
3	2	an32s 650	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → 𝐵 ∈ ℝ)
4	3	fmpttd 6873	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵):𝑍⟶ℝ)
5	4	frnd 6515	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ran (𝑛 ∈ 𝑍 ↦ 𝐵) ⊆ ℝ)
6		mbfsup.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℤ)
7		uzid 12252	. . . . . . . . . 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 2924	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
11	10	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑀 ∈ 𝑍)
12		eqid 2821	. . . . . . . 8 ⊢ (𝑛 ∈ 𝑍 ↦ 𝐵) = (𝑛 ∈ 𝑍 ↦ 𝐵)
13	12, 3	dmmptd 6487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → dom (𝑛 ∈ 𝑍 ↦ 𝐵) = 𝑍)
14	11, 13	eleqtrrd 2916	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑀 ∈ dom (𝑛 ∈ 𝑍 ↦ 𝐵))
15	14	ne0d 4300	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → dom (𝑛 ∈ 𝑍 ↦ 𝐵) ≠ ∅)
16		dm0rn0 5789	. . . . . 6 ⊢ (dom (𝑛 ∈ 𝑍 ↦ 𝐵) = ∅ ↔ ran (𝑛 ∈ 𝑍 ↦ 𝐵) = ∅)
17	16	necon3bii 3068	. . . . 5 ⊢ (dom (𝑛 ∈ 𝑍 ↦ 𝐵) ≠ ∅ ↔ ran (𝑛 ∈ 𝑍 ↦ 𝐵) ≠ ∅)
18	15, 17	sylib 220	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ran (𝑛 ∈ 𝑍 ↦ 𝐵) ≠ ∅)
19		mbfsup.6	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝐵 ≤ 𝑦)
20	4	ffnd 6509	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵) Fn 𝑍)
21		breq1 5061	. . . . . . . . 9 ⊢ (𝑧 = ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) → (𝑧 ≤ 𝑦 ↔ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) ≤ 𝑦))
22	21	ralrn 6848	. . . . . . . 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 6675	. . . . . . . . . 10 ⊢ Ⅎ𝑛((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚)
25		nfcv 2977	. . . . . . . . . 10 ⊢ Ⅎ𝑛 ≤
26		nfcv 2977	. . . . . . . . . 10 ⊢ Ⅎ𝑛𝑦
27	24, 25, 26	nfbr 5105	. . . . . . . . 9 ⊢ Ⅎ𝑛((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) ≤ 𝑦
28		nfv 1911	. . . . . . . . 9 ⊢ Ⅎ𝑚((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) ≤ 𝑦
29		fveq2 6664	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) = ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛))
30	29	breq1d 5068	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) ≤ 𝑦 ↔ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) ≤ 𝑦))
31	27, 28, 30	cbvralw 3441	. . . . . . . 8 ⊢ (∀𝑚 ∈ 𝑍 ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) ≤ 𝑦 ↔ ∀𝑛 ∈ 𝑍 ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) ≤ 𝑦)
32		simpr 487	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍)
33	12	fvmpt2 6773	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝐵 ∈ ℝ) → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = 𝐵)
34	32, 3, 33	syl2anc 586	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = 𝐵)
35	34	breq1d 5068	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → (((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) ≤ 𝑦 ↔ 𝐵 ≤ 𝑦))
36	35	ralbidva 3196	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑛 ∈ 𝑍 ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) ≤ 𝑦 ↔ ∀𝑛 ∈ 𝑍 𝐵 ≤ 𝑦))
37	31, 36	syl5bb 285	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑚 ∈ 𝑍 ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) ≤ 𝑦 ↔ ∀𝑛 ∈ 𝑍 𝐵 ≤ 𝑦))
38	23, 37	bitrd 281	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑧 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)𝑧 ≤ 𝑦 ↔ ∀𝑛 ∈ 𝑍 𝐵 ≤ 𝑦))
39	38	rexbidv 3297	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)𝑧 ≤ 𝑦 ↔ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝐵 ≤ 𝑦))
40	19, 39	mpbird 259	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)𝑧 ≤ 𝑦)
41	5, 18, 40	suprcld 11598	. . 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 6872	. 2 ⊢ (𝜑 → 𝐺:𝐴⟶ℝ)
44		simpr 487	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
45		ltso 10715	. . . . . . . . . . . . . 14 ⊢ < Or ℝ
46	45	supex 8921	. . . . . . . . . . . . 13 ⊢ sup(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ) ∈ V
47	42	fvmpt2 6773	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ∧ sup(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ) ∈ V) → (𝐺‘𝑥) = sup(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ))
48	44, 46, 47	sylancl 588	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = sup(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ))
49	48	breq2d 5070	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (𝑡 < (𝐺‘𝑥) ↔ 𝑡 < sup(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < )))
50	5, 18, 40	3jca 1124	. . . . . . . . . . . . 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 713	. . . . . . . . . . . 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 767	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝑡 ∈ ℝ)
53		suprlub 11599	. . . . . . . . . . . 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 586	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (𝑡 < sup(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ) ↔ ∃𝑧 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)𝑡 < 𝑧))
55	20	adantlr 713	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵) Fn 𝑍)
56		breq2 5062	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) → (𝑡 < 𝑧 ↔ 𝑡 < ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚)))
57	56	rexrn 6847	. . . . . . . . . . . . 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 2977	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛𝑡
60		nfcv 2977	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛 <
61	59, 60, 24	nfbr 5105	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛 𝑡 < ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚)
62		nfv 1911	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑚 𝑡 < ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛)
63	29	breq2d 5070	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → (𝑡 < ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) ↔ 𝑡 < ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛)))
64	61, 62, 63	cbvrexw 3442	. . . . . . . . . . . . 13 ⊢ (∃𝑚 ∈ 𝑍 𝑡 < ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) ↔ ∃𝑛 ∈ 𝑍 𝑡 < ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛))
65	12	fvmpt2i 6772	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ 𝑍 → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = ( I ‘𝐵))
66		eqid 2821	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
67	66	fvmpt2i 6772	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = ( I ‘𝐵))
68	67	adantl 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = ( I ‘𝐵))
69	68	eqcomd 2827	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ( I ‘𝐵) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
70	65, 69	sylan9eqr 2878	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
71	70	breq2d 5070	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → (𝑡 < ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) ↔ 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)))
72	71	rexbidva 3296	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∃𝑛 ∈ 𝑍 𝑡 < ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) ↔ ∃𝑛 ∈ 𝑍 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)))
73	72	adantlr 713	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (∃𝑛 ∈ 𝑍 𝑡 < ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) ↔ ∃𝑛 ∈ 𝑍 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)))
74	64, 73	syl5bb 285	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (∃𝑚 ∈ 𝑍 𝑡 < ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) ↔ ∃𝑛 ∈ 𝑍 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)))
75	58, 74	bitrd 281	. . . . . . . . . . 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 307	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (𝑡 < (𝐺‘𝑥) ↔ ∃𝑛 ∈ 𝑍 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)))
77	76	ralrimiva 3182	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → ∀𝑥 ∈ 𝐴 (𝑡 < (𝐺‘𝑥) ↔ ∃𝑛 ∈ 𝑍 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)))
78		nfv 1911	. . . . . . . . . 10 ⊢ Ⅎ𝑧(𝑡 < (𝐺‘𝑥) ↔ ∃𝑛 ∈ 𝑍 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
79		nfcv 2977	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝑡
80		nfcv 2977	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥 <
81		nfmpt1 5156	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ))
82	42, 81	nfcxfr 2975	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥𝐺
83		nfcv 2977	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥𝑧
84	82, 83	nffv 6674	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝐺‘𝑧)
85	79, 80, 84	nfbr 5105	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 𝑡 < (𝐺‘𝑧)
86		nfcv 2977	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝑍
87		nffvmpt1 6675	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧)
88	79, 80, 87	nfbr 5105	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧)
89	86, 88	nfrex 3309	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∃𝑛 ∈ 𝑍 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧)
90	85, 89	nfbi 1900	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑡 < (𝐺‘𝑧) ↔ ∃𝑛 ∈ 𝑍 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧))
91		fveq2 6664	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝐺‘𝑥) = (𝐺‘𝑧))
92	91	breq2d 5070	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝑡 < (𝐺‘𝑥) ↔ 𝑡 < (𝐺‘𝑧)))
93		fveq2 6664	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧))
94	93	breq2d 5070	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ↔ 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧)))
95	94	rexbidv 3297	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (∃𝑛 ∈ 𝑍 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ↔ ∃𝑛 ∈ 𝑍 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧)))
96	92, 95	bibi12d 348	. . . . . . . . . 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 3441	. . . . . . . . 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 220	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → ∀𝑧 ∈ 𝐴 (𝑡 < (𝐺‘𝑧) ↔ ∃𝑛 ∈ 𝑍 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧)))
99	98	r19.21bi 3208	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑧 ∈ 𝐴) → (𝑡 < (𝐺‘𝑧) ↔ ∃𝑛 ∈ 𝑍 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧)))
100	43	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → 𝐺:𝐴⟶ℝ)
101	100	ffvelrnda 6845	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑧 ∈ 𝐴) → (𝐺‘𝑧) ∈ ℝ)
102		rexr 10681	. . . . . . . . . 10 ⊢ (𝑡 ∈ ℝ → 𝑡 ∈ ℝ^*)
103	102	ad2antlr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑧 ∈ 𝐴) → 𝑡 ∈ ℝ^*)
104		elioopnf 12825	. . . . . . . . 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 705	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑧 ∈ 𝐴) → ((𝐺‘𝑧) ∈ (𝑡(,)+∞) ↔ 𝑡 < (𝐺‘𝑧)))
107	103	adantr 483	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑧 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → 𝑡 ∈ ℝ^*)
108		elioopnf 12825	. . . . . . . . . 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 6873	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ)
111	110	ffvelrnda 6845	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ∈ ℝ)
112	111	biantrurd 535	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝐴) → (𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ↔ (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ∈ ℝ ∧ 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧))))
113	112	an32s 650	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → (𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ↔ (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ∈ ℝ ∧ 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧))))
114	113	adantllr 717	. . . . . . . . 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 284	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑧 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ∈ (𝑡(,)+∞) ↔ 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧)))
116	115	rexbidva 3296	. . . . . . 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 313	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑧 ∈ 𝐴) → ((𝐺‘𝑧) ∈ (𝑡(,)+∞) ↔ ∃𝑛 ∈ 𝑍 ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ∈ (𝑡(,)+∞)))
118	117	pm5.32da 581	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → ((𝑧 ∈ 𝐴 ∧ (𝐺‘𝑧) ∈ (𝑡(,)+∞)) ↔ (𝑧 ∈ 𝐴 ∧ ∃𝑛 ∈ 𝑍 ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ∈ (𝑡(,)+∞))))
119	43	ffnd 6509	. . . . . . 7 ⊢ (𝜑 → 𝐺 Fn 𝐴)
120	119	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → 𝐺 Fn 𝐴)
121		elpreima 6822	. . . . . 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 4915	. . . . . 6 ⊢ (𝑧 ∈ ∪ 𝑛 ∈ 𝑍 (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑡(,)+∞)) ↔ ∃𝑛 ∈ 𝑍 𝑧 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑡(,)+∞)))
124	110	ffnd 6509	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
125		elpreima 6822	. . . . . . . . . 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 3296	. . . . . . . 8 ⊢ (𝜑 → (∃𝑛 ∈ 𝑍 𝑧 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑡(,)+∞)) ↔ ∃𝑛 ∈ 𝑍 (𝑧 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ∈ (𝑡(,)+∞))))
128	127	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → (∃𝑛 ∈ 𝑍 𝑧 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑡(,)+∞)) ↔ ∃𝑛 ∈ 𝑍 (𝑧 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ∈ (𝑡(,)+∞))))
129		r19.42v 3350	. . . . . . 7 ⊢ (∃𝑛 ∈ 𝑍 (𝑧 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ∈ (𝑡(,)+∞)) ↔ (𝑧 ∈ 𝐴 ∧ ∃𝑛 ∈ 𝑍 ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ∈ (𝑡(,)+∞)))
130	128, 129	syl6bb 289	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → (∃𝑛 ∈ 𝑍 𝑧 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑡(,)+∞)) ↔ (𝑧 ∈ 𝐴 ∧ ∃𝑛 ∈ 𝑍 ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ∈ (𝑡(,)+∞))))
131	123, 130	syl5bb 285	. . . . 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 313	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → (𝑧 ∈ (^◡𝐺 “ (𝑡(,)+∞)) ↔ 𝑧 ∈ ∪ 𝑛 ∈ 𝑍 (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑡(,)+∞))))
133	132	eqrdv 2819	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → (^◡𝐺 “ (𝑡(,)+∞)) = ∪ 𝑛 ∈ 𝑍 (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑡(,)+∞)))
134		zex 11984	. . . . . . 7 ⊢ ℤ ∈ V
135		uzssz 12258	. . . . . . 7 ⊢ (ℤ_≥‘𝑀) ⊆ ℤ
136		ssdomg 8549	. . . . . . 7 ⊢ (ℤ ∈ V → ((ℤ_≥‘𝑀) ⊆ ℤ → (ℤ_≥‘𝑀) ≼ ℤ))
137	134, 135, 136	mp2 9	. . . . . 6 ⊢ (ℤ_≥‘𝑀) ≼ ℤ
138	9, 137	eqbrtri 5079	. . . . 5 ⊢ 𝑍 ≼ ℤ
139		znnen 15559	. . . . 5 ⊢ ℤ ≈ ℕ
140		domentr 8562	. . . . 5 ⊢ ((𝑍 ≼ ℤ ∧ ℤ ≈ ℕ) → 𝑍 ≼ ℕ)
141	138, 139, 140	mp2an 690	. . . 4 ⊢ 𝑍 ≼ ℕ
142		mbfsup.4	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
143		mbfima 24225	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ) → (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑡(,)+∞)) ∈ dom vol)
144	142, 110, 143	syl2anc 586	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑡(,)+∞)) ∈ dom vol)
145	144	ralrimiva 3182	. . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ 𝑍 (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑡(,)+∞)) ∈ dom vol)
146	145	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → ∀𝑛 ∈ 𝑍 (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑡(,)+∞)) ∈ dom vol)
147		iunmbl2 24152	. . . 4 ⊢ ((𝑍 ≼ ℕ ∧ ∀𝑛 ∈ 𝑍 (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑡(,)+∞)) ∈ dom vol) → ∪ 𝑛 ∈ 𝑍 (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑡(,)+∞)) ∈ dom vol)
148	141, 146, 147	sylancr 589	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → ∪ 𝑛 ∈ 𝑍 (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑡(,)+∞)) ∈ dom vol)
149	133, 148	eqeltrd 2913	. 2 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → (^◡𝐺 “ (𝑡(,)+∞)) ∈ dom vol)
150	43, 149	ismbf3d 24249	1 ⊢ (𝜑 → 𝐺 ∈ MblFn)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 Vcvv 3494 ⊆ wss 3935 ∅c0 4290 ∪ ciun 4911 class class class wbr 5058 ↦ cmpt 5138 I cid 5453 ^◡ccnv 5548 dom cdm 5549 ran crn 5550 “ cima 5552 Fn wfn 6344 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 ≈ cen 8500 ≼ cdom 8501 supcsup 8898 ℝcr 10530 +∞cpnf 10666 ℝ^*cxr 10668 < clt 10669 ≤ cle 10670 ℕcn 11632 ℤcz 11975 ℤ_≥cuz 12237 (,)cioo 12732 volcvol 24058 MblFncmbf 24209

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 ax-cc 9851 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-disj 5024 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-omul 8101 df-er 8283 df-map 8402 df-pm 8403 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-inf 8901 df-oi 8968 df-dju 9324 df-card 9362 df-acn 9365 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-q 12343 df-rp 12384 df-xadd 12502 df-ioo 12736 df-ioc 12737 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-fl 13156 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-rlim 14840 df-sum 15037 df-xmet 20532 df-met 20533 df-ovol 24059 df-vol 24060 df-mbf 24214

This theorem is referenced by: mbfinf 24260 mbflimsup 24261

W3C validator