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Theorem mbfsup 25631

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 653	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → 𝐵 ∈ ℝ)
4	3	fmpttd 7067	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵):𝑍⟶ℝ)
5	4	frnd 6676	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ran (𝑛 ∈ 𝑍 ↦ 𝐵) ⊆ ℝ)
6		mbfsup.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℤ)
7		uzid 12803	. . . . . . . . . 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 2847	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
11	10	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑀 ∈ 𝑍)
12		eqid 2736	. . . . . . . 8 ⊢ (𝑛 ∈ 𝑍 ↦ 𝐵) = (𝑛 ∈ 𝑍 ↦ 𝐵)
13	12, 3	dmmptd 6643	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → dom (𝑛 ∈ 𝑍 ↦ 𝐵) = 𝑍)
14	11, 13	eleqtrrd 2839	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑀 ∈ dom (𝑛 ∈ 𝑍 ↦ 𝐵))
15	14	ne0d 4282	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → dom (𝑛 ∈ 𝑍 ↦ 𝐵) ≠ ∅)
16		dm0rn0 5879	. . . . . 6 ⊢ (dom (𝑛 ∈ 𝑍 ↦ 𝐵) = ∅ ↔ ran (𝑛 ∈ 𝑍 ↦ 𝐵) = ∅)
17	16	necon3bii 2984	. . . . 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 6669	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵) Fn 𝑍)
21		breq1 5088	. . . . . . . . 9 ⊢ (𝑧 = ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) → (𝑧 ≤ 𝑦 ↔ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) ≤ 𝑦))
22	21	ralrn 7040	. . . . . . . 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 6851	. . . . . . . . . 10 ⊢ Ⅎ𝑛((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚)
25		nfcv 2898	. . . . . . . . . 10 ⊢ Ⅎ𝑛 ≤
26		nfcv 2898	. . . . . . . . . 10 ⊢ Ⅎ𝑛𝑦
27	24, 25, 26	nfbr 5132	. . . . . . . . 9 ⊢ Ⅎ𝑛((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) ≤ 𝑦
28		nfv 1916	. . . . . . . . 9 ⊢ Ⅎ𝑚((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) ≤ 𝑦
29		fveq2 6840	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) = ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛))
30	29	breq1d 5095	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) ≤ 𝑦 ↔ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) ≤ 𝑦))
31	27, 28, 30	cbvralw 3279	. . . . . . . 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 6959	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝐵 ∈ ℝ) → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = 𝐵)
34	32, 3, 33	syl2anc 585	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = 𝐵)
35	34	breq1d 5095	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → (((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) ≤ 𝑦 ↔ 𝐵 ≤ 𝑦))
36	35	ralbidva 3158	. . . . . . . 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 3161	. . . . 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 12119	. . 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 7066	. 2 ⊢ (𝜑 → 𝐺:𝐴⟶ℝ)
44		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
45		ltso 11226	. . . . . . . . . . . . . 14 ⊢ < Or ℝ
46	45	supex 9377	. . . . . . . . . . . . 13 ⊢ sup(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ) ∈ V
47	42	fvmpt2 6959	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ∧ sup(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ) ∈ V) → (𝐺‘𝑥) = sup(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ))
48	44, 46, 47	sylancl 587	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = sup(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ))
49	48	breq2d 5097	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (𝑡 < (𝐺‘𝑥) ↔ 𝑡 < sup(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < )))
50	5, 18, 40	3jca 1129	. . . . . . . . . . . . 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 716	. . . . . . . . . . . 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 769	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝑡 ∈ ℝ)
53		suprlub 12120	. . . . . . . . . . . 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 585	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (𝑡 < sup(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ) ↔ ∃𝑧 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)𝑡 < 𝑧))
55	20	adantlr 716	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵) Fn 𝑍)
56		breq2 5089	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) → (𝑡 < 𝑧 ↔ 𝑡 < ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚)))
57	56	rexrn 7039	. . . . . . . . . . . . 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 2898	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛𝑡
60		nfcv 2898	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛 <
61	59, 60, 24	nfbr 5132	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛 𝑡 < ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚)
62		nfv 1916	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑚 𝑡 < ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛)
63	29	breq2d 5097	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → (𝑡 < ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) ↔ 𝑡 < ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛)))
64	61, 62, 63	cbvrexw 3280	. . . . . . . . . . . . 13 ⊢ (∃𝑚 ∈ 𝑍 𝑡 < ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) ↔ ∃𝑛 ∈ 𝑍 𝑡 < ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛))
65	12	fvmpt2i 6958	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ 𝑍 → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = ( I ‘𝐵))
66		eqid 2736	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
67	66	fvmpt2i 6958	. . . . . . . . . . . . . . . . . . 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 2742	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ( I ‘𝐵) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
70	65, 69	sylan9eqr 2793	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
71	70	breq2d 5097	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → (𝑡 < ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) ↔ 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)))
72	71	rexbidva 3159	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∃𝑛 ∈ 𝑍 𝑡 < ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) ↔ ∃𝑛 ∈ 𝑍 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)))
73	72	adantlr 716	. . . . . . . . . . . . 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 3129	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → ∀𝑥 ∈ 𝐴 (𝑡 < (𝐺‘𝑥) ↔ ∃𝑛 ∈ 𝑍 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)))
78		nfv 1916	. . . . . . . . . 10 ⊢ Ⅎ𝑧(𝑡 < (𝐺‘𝑥) ↔ ∃𝑛 ∈ 𝑍 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
79		nfcv 2898	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝑡
80		nfcv 2898	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥 <
81		nfmpt1 5184	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ))
82	42, 81	nfcxfr 2896	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥𝐺
83		nfcv 2898	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥𝑧
84	82, 83	nffv 6850	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝐺‘𝑧)
85	79, 80, 84	nfbr 5132	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 𝑡 < (𝐺‘𝑧)
86		nfcv 2898	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝑍
87		nffvmpt1 6851	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧)
88	79, 80, 87	nfbr 5132	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧)
89	86, 88	nfrexw 3285	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∃𝑛 ∈ 𝑍 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧)
90	85, 89	nfbi 1905	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑡 < (𝐺‘𝑧) ↔ ∃𝑛 ∈ 𝑍 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧))
91		fveq2 6840	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝐺‘𝑥) = (𝐺‘𝑧))
92	91	breq2d 5097	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝑡 < (𝐺‘𝑥) ↔ 𝑡 < (𝐺‘𝑧)))
93		fveq2 6840	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧))
94	93	breq2d 5097	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ↔ 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧)))
95	94	rexbidv 3161	. . . . . . . . . . 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 3279	. . . . . . . . 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 3229	. . . . . . 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 7036	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑧 ∈ 𝐴) → (𝐺‘𝑧) ∈ ℝ)
102		rexr 11191	. . . . . . . . . 10 ⊢ (𝑡 ∈ ℝ → 𝑡 ∈ ℝ^*)
103	102	ad2antlr 728	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑧 ∈ 𝐴) → 𝑡 ∈ ℝ^*)
104		elioopnf 13396	. . . . . . . . 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 708	. . . . . . 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 13396	. . . . . . . . . 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 7067	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ)
111	110	ffvelcdmda 7036	. . . . . . . . . . . 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 653	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → (𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ↔ (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ∈ ℝ ∧ 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧))))
114	113	adantllr 720	. . . . . . . . 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 3159	. . . . . . 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 6669	. . . . . . 7 ⊢ (𝜑 → 𝐺 Fn 𝐴)
120	119	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → 𝐺 Fn 𝐴)
121		elpreima 7010	. . . . . 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 4937	. . . . . 6 ⊢ (𝑧 ∈ ∪ 𝑛 ∈ 𝑍 (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑡(,)+∞)) ↔ ∃𝑛 ∈ 𝑍 𝑧 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑡(,)+∞)))
124	110	ffnd 6669	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
125		elpreima 7010	. . . . . . . . . 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 3159	. . . . . . . 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 3169	. . . . . . 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 2734	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → (^◡𝐺 “ (𝑡(,)+∞)) = ∪ 𝑛 ∈ 𝑍 (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑡(,)+∞)))
134		zex 12533	. . . . . . 7 ⊢ ℤ ∈ V
135		uzssz 12809	. . . . . . 7 ⊢ (ℤ_≥‘𝑀) ⊆ ℤ
136		ssdomg 8947	. . . . . . 7 ⊢ (ℤ ∈ V → ((ℤ_≥‘𝑀) ⊆ ℤ → (ℤ_≥‘𝑀) ≼ ℤ))
137	134, 135, 136	mp2 9	. . . . . 6 ⊢ (ℤ_≥‘𝑀) ≼ ℤ
138	9, 137	eqbrtri 5106	. . . . 5 ⊢ 𝑍 ≼ ℤ
139		znnen 16179	. . . . 5 ⊢ ℤ ≈ ℕ
140		domentr 8960	. . . . 5 ⊢ ((𝑍 ≼ ℤ ∧ ℤ ≈ ℕ) → 𝑍 ≼ ℕ)
141	138, 139, 140	mp2an 693	. . . 4 ⊢ 𝑍 ≼ ℕ
142		mbfsup.4	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
143		mbfima 25597	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ) → (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑡(,)+∞)) ∈ dom vol)
144	142, 110, 143	syl2anc 585	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑡(,)+∞)) ∈ dom vol)
145	144	ralrimiva 3129	. . . . 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 25524	. . . 4 ⊢ ((𝑍 ≼ ℕ ∧ ∀𝑛 ∈ 𝑍 (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑡(,)+∞)) ∈ dom vol) → ∪ 𝑛 ∈ 𝑍 (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑡(,)+∞)) ∈ dom vol)
148	141, 146, 147	sylancr 588	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → ∪ 𝑛 ∈ 𝑍 (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑡(,)+∞)) ∈ dom vol)
149	133, 148	eqeltrd 2836	. 2 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → (^◡𝐺 “ (𝑡(,)+∞)) ∈ dom vol)
150	43, 149	ismbf3d 25621	1 ⊢ (𝜑 → 𝐺 ∈ MblFn)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2932 ∀wral 3051 ∃wrex 3061 Vcvv 3429 ⊆ wss 3889 ∅c0 4273 ∪ ciun 4933 class class class wbr 5085 ↦ cmpt 5166 I cid 5525 ^◡ccnv 5630 dom cdm 5631 ran crn 5632 “ cima 5634 Fn wfn 6493 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 ≈ cen 8890 ≼ cdom 8891 supcsup 9353 ℝcr 11037 +∞cpnf 11176 ℝ^*cxr 11178 < clt 11179 ≤ cle 11180 ℕcn 12174 ℤcz 12524 ℤ_≥cuz 12788 (,)cioo 13298 volcvol 25430 MblFncmbf 25581

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cc 10357 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-disj 5053 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-oadd 8409 df-omul 8410 df-er 8643 df-map 8775 df-pm 8776 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-sup 9355 df-inf 9356 df-oi 9425 df-dju 9825 df-card 9863 df-acn 9866 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-n0 12438 df-z 12525 df-uz 12789 df-q 12899 df-rp 12943 df-xadd 13064 df-ioo 13302 df-ioc 13303 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-seq 13964 df-exp 14024 df-hash 14293 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-clim 15450 df-rlim 15451 df-sum 15649 df-xmet 21345 df-met 21346 df-ovol 25431 df-vol 25432 df-mbf 25586

This theorem is referenced by: mbfinf 25632 mbflimsup 25633

W3C validator