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Theorem mbfsup 25181

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 469	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
3	2	an32s 651	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → 𝐵 ∈ ℝ)
4	3	fmpttd 7115	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵):𝑍⟶ℝ)
5	4	frnd 6726	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ran (𝑛 ∈ 𝑍 ↦ 𝐵) ⊆ ℝ)
6		mbfsup.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℤ)
7		uzid 12837	. . . . . . . . . 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 2845	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
11	10	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑀 ∈ 𝑍)
12		eqid 2733	. . . . . . . 8 ⊢ (𝑛 ∈ 𝑍 ↦ 𝐵) = (𝑛 ∈ 𝑍 ↦ 𝐵)
13	12, 3	dmmptd 6696	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → dom (𝑛 ∈ 𝑍 ↦ 𝐵) = 𝑍)
14	11, 13	eleqtrrd 2837	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑀 ∈ dom (𝑛 ∈ 𝑍 ↦ 𝐵))
15	14	ne0d 4336	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → dom (𝑛 ∈ 𝑍 ↦ 𝐵) ≠ ∅)
16		dm0rn0 5925	. . . . . 6 ⊢ (dom (𝑛 ∈ 𝑍 ↦ 𝐵) = ∅ ↔ ran (𝑛 ∈ 𝑍 ↦ 𝐵) = ∅)
17	16	necon3bii 2994	. . . . 5 ⊢ (dom (𝑛 ∈ 𝑍 ↦ 𝐵) ≠ ∅ ↔ ran (𝑛 ∈ 𝑍 ↦ 𝐵) ≠ ∅)
18	15, 17	sylib 217	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ran (𝑛 ∈ 𝑍 ↦ 𝐵) ≠ ∅)
19		mbfsup.6	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝐵 ≤ 𝑦)
20	4	ffnd 6719	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵) Fn 𝑍)
21		breq1 5152	. . . . . . . . 9 ⊢ (𝑧 = ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) → (𝑧 ≤ 𝑦 ↔ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) ≤ 𝑦))
22	21	ralrn 7090	. . . . . . . 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 6903	. . . . . . . . . 10 ⊢ Ⅎ𝑛((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚)
25		nfcv 2904	. . . . . . . . . 10 ⊢ Ⅎ𝑛 ≤
26		nfcv 2904	. . . . . . . . . 10 ⊢ Ⅎ𝑛𝑦
27	24, 25, 26	nfbr 5196	. . . . . . . . 9 ⊢ Ⅎ𝑛((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) ≤ 𝑦
28		nfv 1918	. . . . . . . . 9 ⊢ Ⅎ𝑚((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) ≤ 𝑦
29		fveq2 6892	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) = ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛))
30	29	breq1d 5159	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) ≤ 𝑦 ↔ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) ≤ 𝑦))
31	27, 28, 30	cbvralw 3304	. . . . . . . 8 ⊢ (∀𝑚 ∈ 𝑍 ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) ≤ 𝑦 ↔ ∀𝑛 ∈ 𝑍 ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) ≤ 𝑦)
32		simpr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍)
33	12	fvmpt2 7010	. . . . . . . . . . 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 5159	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → (((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) ≤ 𝑦 ↔ 𝐵 ≤ 𝑦))
36	35	ralbidva 3176	. . . . . . . 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 3179	. . . . 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 12177	. . 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 7114	. 2 ⊢ (𝜑 → 𝐺:𝐴⟶ℝ)
44		simpr 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
45		ltso 11294	. . . . . . . . . . . . . 14 ⊢ < Or ℝ
46	45	supex 9458	. . . . . . . . . . . . 13 ⊢ sup(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ) ∈ V
47	42	fvmpt2 7010	. . . . . . . . . . . . 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 5161	. . . . . . . . . . 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 714	. . . . . . . . . . . 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 12178	. . . . . . . . . . . 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 714	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵) Fn 𝑍)
56		breq2 5153	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) → (𝑡 < 𝑧 ↔ 𝑡 < ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚)))
57	56	rexrn 7089	. . . . . . . . . . . . 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 2904	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛𝑡
60		nfcv 2904	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛 <
61	59, 60, 24	nfbr 5196	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛 𝑡 < ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚)
62		nfv 1918	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑚 𝑡 < ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛)
63	29	breq2d 5161	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → (𝑡 < ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) ↔ 𝑡 < ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛)))
64	61, 62, 63	cbvrexw 3305	. . . . . . . . . . . . 13 ⊢ (∃𝑚 ∈ 𝑍 𝑡 < ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) ↔ ∃𝑛 ∈ 𝑍 𝑡 < ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛))
65	12	fvmpt2i 7009	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ 𝑍 → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = ( I ‘𝐵))
66		eqid 2733	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
67	66	fvmpt2i 7009	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = ( I ‘𝐵))
68	67	adantl 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = ( I ‘𝐵))
69	68	eqcomd 2739	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ( I ‘𝐵) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
70	65, 69	sylan9eqr 2795	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
71	70	breq2d 5161	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → (𝑡 < ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) ↔ 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)))
72	71	rexbidva 3177	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∃𝑛 ∈ 𝑍 𝑡 < ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) ↔ ∃𝑛 ∈ 𝑍 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)))
73	72	adantlr 714	. . . . . . . . . . . . 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 3147	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → ∀𝑥 ∈ 𝐴 (𝑡 < (𝐺‘𝑥) ↔ ∃𝑛 ∈ 𝑍 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)))
78		nfv 1918	. . . . . . . . . 10 ⊢ Ⅎ𝑧(𝑡 < (𝐺‘𝑥) ↔ ∃𝑛 ∈ 𝑍 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
79		nfcv 2904	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝑡
80		nfcv 2904	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥 <
81		nfmpt1 5257	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ))
82	42, 81	nfcxfr 2902	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥𝐺
83		nfcv 2904	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥𝑧
84	82, 83	nffv 6902	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝐺‘𝑧)
85	79, 80, 84	nfbr 5196	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 𝑡 < (𝐺‘𝑧)
86		nfcv 2904	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝑍
87		nffvmpt1 6903	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧)
88	79, 80, 87	nfbr 5196	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧)
89	86, 88	nfrexw 3311	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∃𝑛 ∈ 𝑍 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧)
90	85, 89	nfbi 1907	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑡 < (𝐺‘𝑧) ↔ ∃𝑛 ∈ 𝑍 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧))
91		fveq2 6892	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝐺‘𝑥) = (𝐺‘𝑧))
92	91	breq2d 5161	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝑡 < (𝐺‘𝑥) ↔ 𝑡 < (𝐺‘𝑧)))
93		fveq2 6892	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧))
94	93	breq2d 5161	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ↔ 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧)))
95	94	rexbidv 3179	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (∃𝑛 ∈ 𝑍 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ↔ ∃𝑛 ∈ 𝑍 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧)))
96	92, 95	bibi12d 346	. . . . . . . . . 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 3304	. . . . . . . . 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 217	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → ∀𝑧 ∈ 𝐴 (𝑡 < (𝐺‘𝑧) ↔ ∃𝑛 ∈ 𝑍 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧)))
99	98	r19.21bi 3249	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑧 ∈ 𝐴) → (𝑡 < (𝐺‘𝑧) ↔ ∃𝑛 ∈ 𝑍 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧)))
100	43	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → 𝐺:𝐴⟶ℝ)
101	100	ffvelcdmda 7087	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑧 ∈ 𝐴) → (𝐺‘𝑧) ∈ ℝ)
102		rexr 11260	. . . . . . . . . 10 ⊢ (𝑡 ∈ ℝ → 𝑡 ∈ ℝ^*)
103	102	ad2antlr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑧 ∈ 𝐴) → 𝑡 ∈ ℝ^*)
104		elioopnf 13420	. . . . . . . . 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 706	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑧 ∈ 𝐴) → ((𝐺‘𝑧) ∈ (𝑡(,)+∞) ↔ 𝑡 < (𝐺‘𝑧)))
107	103	adantr 482	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑧 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → 𝑡 ∈ ℝ^*)
108		elioopnf 13420	. . . . . . . . . 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 7115	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ)
111	110	ffvelcdmda 7087	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ∈ ℝ)
112	111	biantrurd 534	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝐴) → (𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ↔ (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ∈ ℝ ∧ 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧))))
113	112	an32s 651	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → (𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ↔ (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ∈ ℝ ∧ 𝑡 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧))))
114	113	adantllr 718	. . . . . . . . 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 3177	. . . . . . 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 580	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → ((𝑧 ∈ 𝐴 ∧ (𝐺‘𝑧) ∈ (𝑡(,)+∞)) ↔ (𝑧 ∈ 𝐴 ∧ ∃𝑛 ∈ 𝑍 ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ∈ (𝑡(,)+∞))))
119	43	ffnd 6719	. . . . . . 7 ⊢ (𝜑 → 𝐺 Fn 𝐴)
120	119	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → 𝐺 Fn 𝐴)
121		elpreima 7060	. . . . . 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 5002	. . . . . 6 ⊢ (𝑧 ∈ ∪ 𝑛 ∈ 𝑍 (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑡(,)+∞)) ↔ ∃𝑛 ∈ 𝑍 𝑧 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑡(,)+∞)))
124	110	ffnd 6719	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
125		elpreima 7060	. . . . . . . . . 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 3177	. . . . . . . 8 ⊢ (𝜑 → (∃𝑛 ∈ 𝑍 𝑧 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑡(,)+∞)) ↔ ∃𝑛 ∈ 𝑍 (𝑧 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ∈ (𝑡(,)+∞))))
128	127	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → (∃𝑛 ∈ 𝑍 𝑧 ∈ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑡(,)+∞)) ↔ ∃𝑛 ∈ 𝑍 (𝑧 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ∈ (𝑡(,)+∞))))
129		r19.42v 3191	. . . . . . 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 2731	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → (^◡𝐺 “ (𝑡(,)+∞)) = ∪ 𝑛 ∈ 𝑍 (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑡(,)+∞)))
134		zex 12567	. . . . . . 7 ⊢ ℤ ∈ V
135		uzssz 12843	. . . . . . 7 ⊢ (ℤ_≥‘𝑀) ⊆ ℤ
136		ssdomg 8996	. . . . . . 7 ⊢ (ℤ ∈ V → ((ℤ_≥‘𝑀) ⊆ ℤ → (ℤ_≥‘𝑀) ≼ ℤ))
137	134, 135, 136	mp2 9	. . . . . 6 ⊢ (ℤ_≥‘𝑀) ≼ ℤ
138	9, 137	eqbrtri 5170	. . . . 5 ⊢ 𝑍 ≼ ℤ
139		znnen 16155	. . . . 5 ⊢ ℤ ≈ ℕ
140		domentr 9009	. . . . 5 ⊢ ((𝑍 ≼ ℤ ∧ ℤ ≈ ℕ) → 𝑍 ≼ ℕ)
141	138, 139, 140	mp2an 691	. . . 4 ⊢ 𝑍 ≼ ℕ
142		mbfsup.4	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
143		mbfima 25147	. . . . . . 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 3147	. . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ 𝑍 (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑡(,)+∞)) ∈ dom vol)
146	145	adantr 482	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → ∀𝑛 ∈ 𝑍 (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (𝑡(,)+∞)) ∈ dom vol)
147		iunmbl2 25074	. . . 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 2834	. 2 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → (^◡𝐺 “ (𝑡(,)+∞)) ∈ dom vol)
150	43, 149	ismbf3d 25171	1 ⊢ (𝜑 → 𝐺 ∈ MblFn)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 Vcvv 3475 ⊆ wss 3949 ∅c0 4323 ∪ ciun 4998 class class class wbr 5149 ↦ cmpt 5232 I cid 5574 ^◡ccnv 5676 dom cdm 5677 ran crn 5678 “ cima 5680 Fn wfn 6539 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 ≈ cen 8936 ≼ cdom 8937 supcsup 9435 ℝcr 11109 +∞cpnf 11245 ℝ^*cxr 11247 < clt 11248 ≤ cle 11249 ℕcn 12212 ℤcz 12558 ℤ_≥cuz 12822 (,)cioo 13324 volcvol 24980 MblFncmbf 25131

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cc 10430 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-disj 5115 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-oadd 8470 df-omul 8471 df-er 8703 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-inf 9438 df-oi 9505 df-dju 9896 df-card 9934 df-acn 9937 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-q 12933 df-rp 12975 df-xadd 13093 df-ioo 13328 df-ioc 13329 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-fl 13757 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-rlim 15433 df-sum 15633 df-xmet 20937 df-met 20938 df-ovol 24981 df-vol 24982 df-mbf 25136

This theorem is referenced by: mbfinf 25182 mbflimsup 25183

W3C validator