Theorem cnambfre 38050

Description: A real-valued, a.e. continuous function is measurable. (Contributed by Brendan Leahy, 4-Apr-2018.)

Assertion

Ref	Expression
cnambfre	⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol ∧ (vol*‘(𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹}))) = 0) → 𝐹 ∈ MblFn)

Proof of Theorem cnambfre

Dummy variables 𝑓 𝑏 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶ℝ → 𝐹:𝐴⟶ℝ)
2	1	feqmptd 6899	. . . . . . . . 9 ⊢ (𝐹:𝐴⟶ℝ → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
3	2	cnveqd 5820	. . . . . . . 8 ⊢ (𝐹:𝐴⟶ℝ → ◡𝐹 = ◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
4	3	imaeq1d 6018	. . . . . . 7 ⊢ (𝐹:𝐴⟶ℝ → (◡𝐹 “ 𝑏) = (◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) “ 𝑏))
5	4	ad2antrr 733	. . . . . 6 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol) ∧ 𝑏 ∈ ran (,)) → (◡𝐹 “ 𝑏) = (◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) “ 𝑏))
6		exmid 901	. . . . . . . . . . 11 ⊢ (𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∨ ¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥))
7	6	biantrur 536	. . . . . . . . . 10 ⊢ ((𝐹‘𝑥) ∈ 𝑏 ↔ ((𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∨ ¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)) ∧ (𝐹‘𝑥) ∈ 𝑏))
8		andir 1017	. . . . . . . . . 10 ⊢ (((𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∨ ¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)) ∧ (𝐹‘𝑥) ∈ 𝑏) ↔ ((𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝑏) ∨ (¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝑏)))
9	7, 8	bitri 277	. . . . . . . . 9 ⊢ ((𝐹‘𝑥) ∈ 𝑏 ↔ ((𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝑏) ∨ (¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝑏)))
10		retopbas 24747	. . . . . . . . . . . . . . . . . 18 ⊢ ran (,) ∈ TopBases
11		bastg 22953	. . . . . . . . . . . . . . . . . 18 ⊢ (ran (,) ∈ TopBases → ran (,) ⊆ (topGen‘ran (,)))
12	10, 11	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ran (,) ⊆ (topGen‘ran (,))
13	12	sseli 3913	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ∈ ran (,) → 𝑏 ∈ (topGen‘ran (,)))
14	13	ad2antlr 734	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol) ∧ 𝑏 ∈ ran (,)) ∧ 𝑥 ∈ 𝐴) → 𝑏 ∈ (topGen‘ran (,)))
15		cnpimaex 23243	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ 𝑏 ∈ (topGen‘ran (,)) ∧ (𝐹‘𝑥) ∈ 𝑏) → ∃𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)(𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))
16	15	3com12 1130	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 ∈ (topGen‘ran (,)) ∧ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝑏) → ∃𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)(𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))
17	16	3expa 1125	. . . . . . . . . . . . . . 15 ⊢ (((𝑏 ∈ (topGen‘ran (,)) ∧ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)) ∧ (𝐹‘𝑥) ∈ 𝑏) → ∃𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)(𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))
18	14, 17	sylanl1 687	. . . . . . . . . . . . . 14 ⊢ ((((((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol) ∧ 𝑏 ∈ ran (,)) ∧ 𝑥 ∈ 𝐴) ∧ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)) ∧ (𝐹‘𝑥) ∈ 𝑏) → ∃𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)(𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))
19	18	ex 414	. . . . . . . . . . . . 13 ⊢ (((((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol) ∧ 𝑏 ∈ ran (,)) ∧ 𝑥 ∈ 𝐴) ∧ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)) → ((𝐹‘𝑥) ∈ 𝑏 → ∃𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)(𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)))
20		simprrr 788	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol) ∧ (𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))) → (𝐹 “ 𝑦) ⊆ 𝑏)
21		ffn 6659	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹:𝐴⟶ℝ → 𝐹 Fn 𝐴)
22	21	adantr 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol) → 𝐹 Fn 𝐴)
23		restsspw 17389	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((topGen‘ran (,)) ↾t 𝐴) ⊆ 𝒫 𝐴
24	23	sseli 3913	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) → 𝑦 ∈ 𝒫 𝐴)
25	24	elpwid 4541	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) → 𝑦 ⊆ 𝐴)
26		simpl 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏) → 𝑥 ∈ 𝑦)
27		fnfvima 7181	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑦 ⊆ 𝐴 ∧ 𝑥 ∈ 𝑦) → (𝐹‘𝑥) ∈ (𝐹 “ 𝑦))
28	22, 25, 26, 27	syl3an 1167	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol) ∧ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)) → (𝐹‘𝑥) ∈ (𝐹 “ 𝑦))
29	28	3expb 1127	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol) ∧ (𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))) → (𝐹‘𝑥) ∈ (𝐹 “ 𝑦))
30	20, 29	sseldd 3918	. . . . . . . . . . . . . . 15 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol) ∧ (𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))) → (𝐹‘𝑥) ∈ 𝑏)
31	30	rexlimdvaa 3143	. . . . . . . . . . . . . 14 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol) → (∃𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)(𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏) → (𝐹‘𝑥) ∈ 𝑏))
32	31	ad3antrrr 737	. . . . . . . . . . . . 13 ⊢ (((((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol) ∧ 𝑏 ∈ ran (,)) ∧ 𝑥 ∈ 𝐴) ∧ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)) → (∃𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)(𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏) → (𝐹‘𝑥) ∈ 𝑏))
33	19, 32	impbid 214	. . . . . . . . . . . 12 ⊢ (((((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol) ∧ 𝑏 ∈ ran (,)) ∧ 𝑥 ∈ 𝐴) ∧ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)) → ((𝐹‘𝑥) ∈ 𝑏 ↔ ∃𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)(𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)))
34	33	pm5.32da 585	. . . . . . . . . . 11 ⊢ ((((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol) ∧ 𝑏 ∈ ran (,)) ∧ 𝑥 ∈ 𝐴) → ((𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝑏) ↔ (𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ ∃𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)(𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))))
35		r19.42v 3173	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)(𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)) ↔ (𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ ∃𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)(𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)))
36	34, 35	bitr4di 291	. . . . . . . . . 10 ⊢ ((((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol) ∧ 𝑏 ∈ ran (,)) ∧ 𝑥 ∈ 𝐴) → ((𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝑏) ↔ ∃𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)(𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))))
37	36	orbi1d 923	. . . . . . . . 9 ⊢ ((((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol) ∧ 𝑏 ∈ ran (,)) ∧ 𝑥 ∈ 𝐴) → (((𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝑏) ∨ (¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝑏)) ↔ (∃𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)(𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)) ∨ (¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝑏))))
38	9, 37	bitrid 285	. . . . . . . 8 ⊢ ((((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol) ∧ 𝑏 ∈ ran (,)) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) ∈ 𝑏 ↔ (∃𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)(𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)) ∨ (¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝑏))))
39	38	rabbidva 3399	. . . . . . 7 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol) ∧ 𝑏 ∈ ran (,)) → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ∈ 𝑏} = {𝑥 ∈ 𝐴 ∣ (∃𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)(𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)) ∨ (¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝑏))})
40		eqid 2741	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))
41	40	mptpreima 6193	. . . . . . 7 ⊢ (◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) “ 𝑏) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ∈ 𝑏}
42		unrab 4246	. . . . . . 7 ⊢ ({𝑥 ∈ 𝐴 ∣ ∃𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)(𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))} ∪ {𝑥 ∈ 𝐴 ∣ (¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝑏)}) = {𝑥 ∈ 𝐴 ∣ (∃𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)(𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)) ∨ (¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝑏))}
43	39, 41, 42	3eqtr4g 2801	. . . . . 6 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol) ∧ 𝑏 ∈ ran (,)) → (◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) “ 𝑏) = ({𝑥 ∈ 𝐴 ∣ ∃𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)(𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))} ∪ {𝑥 ∈ 𝐴 ∣ (¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝑏)}))
44	5, 43	eqtrd 2776	. . . . 5 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol) ∧ 𝑏 ∈ ran (,)) → (◡𝐹 “ 𝑏) = ({𝑥 ∈ 𝐴 ∣ ∃𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)(𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))} ∪ {𝑥 ∈ 𝐴 ∣ (¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝑏)}))
45	44	3adantl3 1176	. . . 4 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol ∧ (vol*‘(𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹}))) = 0) ∧ 𝑏 ∈ ran (,)) → (◡𝐹 “ 𝑏) = ({𝑥 ∈ 𝐴 ∣ ∃𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)(𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))} ∪ {𝑥 ∈ 𝐴 ∣ (¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝑏)}))
46		incom 4141	. . . . . . . . 9 ⊢ (∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ∩ {𝑥 ∈ 𝐴 ∣ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)}) = ({𝑥 ∈ 𝐴 ∣ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)} ∩ ∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)})
47		dfin4 4209	. . . . . . . . 9 ⊢ (∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ∩ {𝑥 ∈ 𝐴 ∣ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)}) = (∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ∖ (∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ∖ {𝑥 ∈ 𝐴 ∣ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)}))
48		inrab 4247	. . . . . . . . . . . 12 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)} ∩ {𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)}) = {𝑥 ∈ 𝐴 ∣ (𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))}
49	48	a1i 11	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) → ({𝑥 ∈ 𝐴 ∣ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)} ∩ {𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)}) = {𝑥 ∈ 𝐴 ∣ (𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))})
50	49	iuneq2i 4946	. . . . . . . . . 10 ⊢ ∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)({𝑥 ∈ 𝐴 ∣ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)} ∩ {𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)}) = ∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))}
51		iunin2 5003	. . . . . . . . . 10 ⊢ ∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)({𝑥 ∈ 𝐴 ∣ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)} ∩ {𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)}) = ({𝑥 ∈ 𝐴 ∣ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)} ∩ ∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)})
52		iunrab 4985	. . . . . . . . . 10 ⊢ ∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))} = {𝑥 ∈ 𝐴 ∣ ∃𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)(𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))}
53	50, 51, 52	3eqtr3i 2772	. . . . . . . . 9 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)} ∩ ∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)}) = {𝑥 ∈ 𝐴 ∣ ∃𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)(𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))}
54	46, 47, 53	3eqtr3i 2772	. . . . . . . 8 ⊢ (∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ∖ (∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ∖ {𝑥 ∈ 𝐴 ∣ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)})) = {𝑥 ∈ 𝐴 ∣ ∃𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)(𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))}
55		eqeq2 2753	. . . . . . . . . . . 12 ⊢ (𝑦 = if((𝐹 “ 𝑦) ⊆ 𝑏, 𝑦, ∅) → ({𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} = 𝑦 ↔ {𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} = if((𝐹 “ 𝑦) ⊆ 𝑏, 𝑦, ∅)))
56		eqeq2 2753	. . . . . . . . . . . 12 ⊢ (∅ = if((𝐹 “ 𝑦) ⊆ 𝑏, 𝑦, ∅) → ({𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} = ∅ ↔ {𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} = if((𝐹 “ 𝑦) ⊆ 𝑏, 𝑦, ∅)))
57		simprrl 787	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝐹 “ 𝑦) ⊆ 𝑏) ∧ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))) → 𝑥 ∈ 𝑦)
58	25	adantr 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝐹 “ 𝑦) ⊆ 𝑏) → 𝑦 ⊆ 𝐴)
59	58	sselda 3917	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝐹 “ 𝑦) ⊆ 𝑏) ∧ 𝑥 ∈ 𝑦) → 𝑥 ∈ 𝐴)
60		pm3.22 461	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹 “ 𝑦) ⊆ 𝑏 ∧ 𝑥 ∈ 𝑦) → (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))
61	60	adantll 721	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝐹 “ 𝑦) ⊆ 𝑏) ∧ 𝑥 ∈ 𝑦) → (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))
62	59, 61	jca 517	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝐹 “ 𝑦) ⊆ 𝑏) ∧ 𝑥 ∈ 𝑦) → (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)))
63	57, 62	impbida 807	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝐹 “ 𝑦) ⊆ 𝑏) → ((𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)) ↔ 𝑥 ∈ 𝑦))
64	63	abbidv 2807	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝐹 “ 𝑦) ⊆ 𝑏) → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))} = {𝑥 ∣ 𝑥 ∈ 𝑦})
65		df-rab 3394	. . . . . . . . . . . . 13 ⊢ {𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))}
66		cvjust 2735	. . . . . . . . . . . . 13 ⊢ 𝑦 = {𝑥 ∣ 𝑥 ∈ 𝑦}
67	64, 65, 66	3eqtr4g 2801	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝐹 “ 𝑦) ⊆ 𝑏) → {𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} = 𝑦)
68		simpr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏) → (𝐹 “ 𝑦) ⊆ 𝑏)
69	68	con3i 154	. . . . . . . . . . . . . . 15 ⊢ (¬ (𝐹 “ 𝑦) ⊆ 𝑏 → ¬ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))
70	69	ralrimivw 3137	. . . . . . . . . . . . . 14 ⊢ (¬ (𝐹 “ 𝑦) ⊆ 𝑏 → ∀𝑥 ∈ 𝐴 ¬ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))
71		rabeq0 4319	. . . . . . . . . . . . . 14 ⊢ ({𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))
72	70, 71	sylibr 236	. . . . . . . . . . . . 13 ⊢ (¬ (𝐹 “ 𝑦) ⊆ 𝑏 → {𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} = ∅)
73	72	adantl 483	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ ¬ (𝐹 “ 𝑦) ⊆ 𝑏) → {𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} = ∅)
74	55, 56, 67, 73	ifbothda 4496	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) → {𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} = if((𝐹 “ 𝑦) ⊆ 𝑏, 𝑦, ∅))
75	74	iuneq2i 4946	. . . . . . . . . 10 ⊢ ∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} = ∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)if((𝐹 “ 𝑦) ⊆ 𝑏, 𝑦, ∅)
76		retop 24748	. . . . . . . . . . . . 13 ⊢ (topGen‘ran (,)) ∈ Top
77		resttop 23147	. . . . . . . . . . . . 13 ⊢ (((topGen‘ran (,)) ∈ Top ∧ 𝐴 ∈ dom vol) → ((topGen‘ran (,)) ↾t 𝐴) ∈ Top)
78	76, 77	mpan 697	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom vol → ((topGen‘ran (,)) ↾t 𝐴) ∈ Top)
79		0opn 22891	. . . . . . . . . . . . . . 15 ⊢ (((topGen‘ran (,)) ↾t 𝐴) ∈ Top → ∅ ∈ ((topGen‘ran (,)) ↾t 𝐴))
80	78, 79	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ dom vol → ∅ ∈ ((topGen‘ran (,)) ↾t 𝐴))
81		ifcl 4503	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ ∅ ∈ ((topGen‘ran (,)) ↾t 𝐴)) → if((𝐹 “ 𝑦) ⊆ 𝑏, 𝑦, ∅) ∈ ((topGen‘ran (,)) ↾t 𝐴))
82	81	ancoms 460	. . . . . . . . . . . . . 14 ⊢ ((∅ ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)) → if((𝐹 “ 𝑦) ⊆ 𝑏, 𝑦, ∅) ∈ ((topGen‘ran (,)) ↾t 𝐴))
83	80, 82	sylan 587	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ dom vol ∧ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)) → if((𝐹 “ 𝑦) ⊆ 𝑏, 𝑦, ∅) ∈ ((topGen‘ran (,)) ↾t 𝐴))
84	83	ralrimiva 3133	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom vol → ∀𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)if((𝐹 “ 𝑦) ⊆ 𝑏, 𝑦, ∅) ∈ ((topGen‘ran (,)) ↾t 𝐴))
85		iunopn 22885	. . . . . . . . . . . 12 ⊢ ((((topGen‘ran (,)) ↾t 𝐴) ∈ Top ∧ ∀𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)if((𝐹 “ 𝑦) ⊆ 𝑏, 𝑦, ∅) ∈ ((topGen‘ran (,)) ↾t 𝐴)) → ∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)if((𝐹 “ 𝑦) ⊆ 𝑏, 𝑦, ∅) ∈ ((topGen‘ran (,)) ↾t 𝐴))
86	78, 84, 85	syl2anc 591	. . . . . . . . . . 11 ⊢ (𝐴 ∈ dom vol → ∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)if((𝐹 “ 𝑦) ⊆ 𝑏, 𝑦, ∅) ∈ ((topGen‘ran (,)) ↾t 𝐴))
87		eqid 2741	. . . . . . . . . . . 12 ⊢ ((topGen‘ran (,)) ↾t 𝐴) = ((topGen‘ran (,)) ↾t 𝐴)
88	87	subopnmbl 25593	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ dom vol ∧ ∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)if((𝐹 “ 𝑦) ⊆ 𝑏, 𝑦, ∅) ∈ ((topGen‘ran (,)) ↾t 𝐴)) → ∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)if((𝐹 “ 𝑦) ⊆ 𝑏, 𝑦, ∅) ∈ dom vol)
89	86, 88	mpdan 694	. . . . . . . . . 10 ⊢ (𝐴 ∈ dom vol → ∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)if((𝐹 “ 𝑦) ⊆ 𝑏, 𝑦, ∅) ∈ dom vol)
90	75, 89	eqeltrid 2845	. . . . . . . . 9 ⊢ (𝐴 ∈ dom vol → ∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ∈ dom vol)
91		difss 4069	. . . . . . . . . . . 12 ⊢ (∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ∖ {𝑥 ∈ 𝐴 ∣ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)}) ⊆ ∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)}
92		ssrab2 4014	. . . . . . . . . . . . . 14 ⊢ {𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ⊆ 𝐴
93	92	rgenw 3059	. . . . . . . . . . . . 13 ⊢ ∀𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ⊆ 𝐴
94		iunss 4977	. . . . . . . . . . . . 13 ⊢ (∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ⊆ 𝐴 ↔ ∀𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ⊆ 𝐴)
95	93, 94	mpbir 233	. . . . . . . . . . . 12 ⊢ ∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ⊆ 𝐴
96	91, 95	sstri 3926	. . . . . . . . . . 11 ⊢ (∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ∖ {𝑥 ∈ 𝐴 ∣ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)}) ⊆ 𝐴
97		mblss 25520	. . . . . . . . . . 11 ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ)
98	96, 97	sstrid 3928	. . . . . . . . . 10 ⊢ (𝐴 ∈ dom vol → (∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ∖ {𝑥 ∈ 𝐴 ∣ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)}) ⊆ ℝ)
99		ssdif 4077	. . . . . . . . . . . . . 14 ⊢ (∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ⊆ 𝐴 → (∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ∖ {𝑥 ∈ 𝐴 ∣ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)}) ⊆ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)}))
100	95, 99	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ∖ {𝑥 ∈ 𝐴 ∣ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)}) ⊆ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)})
101		rele 5773	. . . . . . . . . . . . . . . . . . . 20 ⊢ Rel E
102		elrelimasn 6045	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Rel E → (((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∈ ( E “ {𝐹}) ↔ 𝐹 E ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)))
103	101, 102	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∈ ( E “ {𝐹}) ↔ 𝐹 E ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥))
104		fvex 6844	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∈ V
105	104	epeli 5523	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹 E ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ↔ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥))
106	103, 105	bitr2i 278	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ↔ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∈ ( E “ {𝐹}))
107	106	anbi2i 630	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)) ↔ (𝑥 ∈ 𝐴 ∧ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∈ ( E “ {𝐹})))
108		ovex 7393	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℝ ↑m 𝐴) ∈ V
109	108	rabex 5270	. . . . . . . . . . . . . . . . . . . 20 ⊢ {𝑓 ∈ (ℝ ↑m 𝐴) ∣ ∀𝑏 ∈ (topGen‘ran (,))((𝑓‘𝑥) ∈ 𝑏 → ∃𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)(𝑥 ∈ 𝑦 ∧ (𝑓 “ 𝑦) ⊆ 𝑏))} ∈ V
110		eqid 2741	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑓 ∈ (ℝ ↑m 𝐴) ∣ ∀𝑏 ∈ (topGen‘ran (,))((𝑓‘𝑥) ∈ 𝑏 → ∃𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)(𝑥 ∈ 𝑦 ∧ (𝑓 “ 𝑦) ⊆ 𝑏))}) = (𝑥 ∈ 𝐴 ↦ {𝑓 ∈ (ℝ ↑m 𝐴) ∣ ∀𝑏 ∈ (topGen‘ran (,))((𝑓‘𝑥) ∈ 𝑏 → ∃𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)(𝑥 ∈ 𝑦 ∧ (𝑓 “ 𝑦) ⊆ 𝑏))})
111	109, 110	fnmpti 6632	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑓 ∈ (ℝ ↑m 𝐴) ∣ ∀𝑏 ∈ (topGen‘ran (,))((𝑓‘𝑥) ∈ 𝑏 → ∃𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)(𝑥 ∈ 𝑦 ∧ (𝑓 “ 𝑦) ⊆ 𝑏))}) Fn 𝐴
112		retopon 24750	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
113		resttopon 23148	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ 𝐴 ⊆ ℝ) → ((topGen‘ran (,)) ↾t 𝐴) ∈ (TopOn‘𝐴))
114	112, 97, 113	sylancr 594	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ∈ dom vol → ((topGen‘ran (,)) ↾t 𝐴) ∈ (TopOn‘𝐴))
115		cnpfval 23221	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((topGen‘ran (,)) ↾t 𝐴) ∈ (TopOn‘𝐴) ∧ (topGen‘ran (,)) ∈ (TopOn‘ℝ)) → (((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) = (𝑥 ∈ 𝐴 ↦ {𝑓 ∈ (ℝ ↑m 𝐴) ∣ ∀𝑏 ∈ (topGen‘ran (,))((𝑓‘𝑥) ∈ 𝑏 → ∃𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)(𝑥 ∈ 𝑦 ∧ (𝑓 “ 𝑦) ⊆ 𝑏))}))
116	114, 112, 115	sylancl 593	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ∈ dom vol → (((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) = (𝑥 ∈ 𝐴 ↦ {𝑓 ∈ (ℝ ↑m 𝐴) ∣ ∀𝑏 ∈ (topGen‘ran (,))((𝑓‘𝑥) ∈ 𝑏 → ∃𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)(𝑥 ∈ 𝑦 ∧ (𝑓 “ 𝑦) ⊆ 𝑏))}))
117	116	fneq1d 6582	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∈ dom vol → ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) Fn 𝐴 ↔ (𝑥 ∈ 𝐴 ↦ {𝑓 ∈ (ℝ ↑m 𝐴) ∣ ∀𝑏 ∈ (topGen‘ran (,))((𝑓‘𝑥) ∈ 𝑏 → ∃𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)(𝑥 ∈ 𝑦 ∧ (𝑓 “ 𝑦) ⊆ 𝑏))}) Fn 𝐴))
118	111, 117	mpbiri 260	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ dom vol → (((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) Fn 𝐴)
119		elpreima 7003	. . . . . . . . . . . . . . . . . 18 ⊢ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) Fn 𝐴 → (𝑥 ∈ (◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) “ ( E “ {𝐹})) ↔ (𝑥 ∈ 𝐴 ∧ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∈ ( E “ {𝐹}))))
120	118, 119	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ dom vol → (𝑥 ∈ (◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) “ ( E “ {𝐹})) ↔ (𝑥 ∈ 𝐴 ∧ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∈ ( E “ {𝐹}))))
121	107, 120	bitr4id 292	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ dom vol → ((𝑥 ∈ 𝐴 ∧ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)) ↔ 𝑥 ∈ (◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) “ ( E “ {𝐹}))))
122	121	abbidv 2807	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ dom vol → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥))} = {𝑥 ∣ 𝑥 ∈ (◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) “ ( E “ {𝐹}))})
123		df-rab 3394	. . . . . . . . . . . . . . 15 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥))}
124		imaco 6206	. . . . . . . . . . . . . . . 16 ⊢ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹}) = (◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) “ ( E “ {𝐹}))
125		abid2 2878	. . . . . . . . . . . . . . . 16 ⊢ {𝑥 ∣ 𝑥 ∈ (◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) “ ( E “ {𝐹}))} = (◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) “ ( E “ {𝐹}))
126	124, 125	eqtr4i 2767	. . . . . . . . . . . . . . 15 ⊢ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹}) = {𝑥 ∣ 𝑥 ∈ (◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) “ ( E “ {𝐹}))}
127	122, 123, 126	3eqtr4g 2801	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ dom vol → {𝑥 ∈ 𝐴 ∣ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)} = ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹}))
128	127	difeq2d 4060	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ dom vol → (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)}) = (𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹})))
129	100, 128	sseqtrid 3959	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom vol → (∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ∖ {𝑥 ∈ 𝐴 ∣ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)}) ⊆ (𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹})))
130		difss 4069	. . . . . . . . . . . . 13 ⊢ (𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹})) ⊆ 𝐴
131	130, 97	sstrid 3928	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom vol → (𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹})) ⊆ ℝ)
132	129, 131	jca 517	. . . . . . . . . . 11 ⊢ (𝐴 ∈ dom vol → ((∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ∖ {𝑥 ∈ 𝐴 ∣ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)}) ⊆ (𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹})) ∧ (𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹})) ⊆ ℝ))
133		ovolssnul 25476	. . . . . . . . . . . 12 ⊢ (((∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ∖ {𝑥 ∈ 𝐴 ∣ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)}) ⊆ (𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹})) ∧ (𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹})) ⊆ ℝ ∧ (vol*‘(𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹}))) = 0) → (vol*‘(∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ∖ {𝑥 ∈ 𝐴 ∣ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)})) = 0)
134	133	3expa 1125	. . . . . . . . . . 11 ⊢ ((((∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ∖ {𝑥 ∈ 𝐴 ∣ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)}) ⊆ (𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹})) ∧ (𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹})) ⊆ ℝ) ∧ (vol*‘(𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹}))) = 0) → (vol*‘(∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ∖ {𝑥 ∈ 𝐴 ∣ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)})) = 0)
135	132, 134	sylan 587	. . . . . . . . . 10 ⊢ ((𝐴 ∈ dom vol ∧ (vol*‘(𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹}))) = 0) → (vol*‘(∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ∖ {𝑥 ∈ 𝐴 ∣ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)})) = 0)
136		nulmbl 25524	. . . . . . . . . 10 ⊢ (((∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ∖ {𝑥 ∈ 𝐴 ∣ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)}) ⊆ ℝ ∧ (vol*‘(∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ∖ {𝑥 ∈ 𝐴 ∣ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)})) = 0) → (∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ∖ {𝑥 ∈ 𝐴 ∣ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)}) ∈ dom vol)
137	98, 135, 136	syl2an2r 692	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom vol ∧ (vol*‘(𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹}))) = 0) → (∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ∖ {𝑥 ∈ 𝐴 ∣ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)}) ∈ dom vol)
138		difmbl 25532	. . . . . . . . 9 ⊢ ((∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ∈ dom vol ∧ (∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ∖ {𝑥 ∈ 𝐴 ∣ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)}) ∈ dom vol) → (∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ∖ (∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ∖ {𝑥 ∈ 𝐴 ∣ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)})) ∈ dom vol)
139	90, 137, 138	syl2an2r 692	. . . . . . . 8 ⊢ ((𝐴 ∈ dom vol ∧ (vol*‘(𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹}))) = 0) → (∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ∖ (∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ∖ {𝑥 ∈ 𝐴 ∣ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)})) ∈ dom vol)
140	54, 139	eqeltrrid 2846	. . . . . . 7 ⊢ ((𝐴 ∈ dom vol ∧ (vol*‘(𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹}))) = 0) → {𝑥 ∈ 𝐴 ∣ ∃𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)(𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))} ∈ dom vol)
141		ssrab2 4014	. . . . . . . . 9 ⊢ {𝑥 ∈ 𝐴 ∣ (¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝑏)} ⊆ 𝐴
142	141, 97	sstrid 3928	. . . . . . . 8 ⊢ (𝐴 ∈ dom vol → {𝑥 ∈ 𝐴 ∣ (¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝑏)} ⊆ ℝ)
143	124	eleq2i 2833	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹}) ↔ 𝑥 ∈ (◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) “ ( E “ {𝐹})))
144		ibar 534	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ 𝐴 → (((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∈ ( E “ {𝐹}) ↔ (𝑥 ∈ 𝐴 ∧ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∈ ( E “ {𝐹}))))
145	106, 144	bitr2id 286	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ∧ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∈ ( E “ {𝐹})) ↔ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)))
146	120, 145	sylan9bb 515	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ dom vol ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ (◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) “ ( E “ {𝐹})) ↔ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)))
147	143, 146	bitr2id 286	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ dom vol ∧ 𝑥 ∈ 𝐴) → (𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ↔ 𝑥 ∈ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹})))
148	147	notbid 320	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ dom vol ∧ 𝑥 ∈ 𝐴) → (¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ↔ ¬ 𝑥 ∈ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹})))
149	148	biimpd 231	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ dom vol ∧ 𝑥 ∈ 𝐴) → (¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) → ¬ 𝑥 ∈ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹})))
150	149	adantrd 493	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ dom vol ∧ 𝑥 ∈ 𝐴) → ((¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝑏) → ¬ 𝑥 ∈ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹})))
151	150	ss2rabdv 4009	. . . . . . . . . . 11 ⊢ (𝐴 ∈ dom vol → {𝑥 ∈ 𝐴 ∣ (¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝑏)} ⊆ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹})})
152		dfdif2 3894	. . . . . . . . . . 11 ⊢ (𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹})) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹})}
153	151, 152	sseqtrrdi 3958	. . . . . . . . . 10 ⊢ (𝐴 ∈ dom vol → {𝑥 ∈ 𝐴 ∣ (¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝑏)} ⊆ (𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹})))
154	153, 131	jca 517	. . . . . . . . 9 ⊢ (𝐴 ∈ dom vol → ({𝑥 ∈ 𝐴 ∣ (¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝑏)} ⊆ (𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹})) ∧ (𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹})) ⊆ ℝ))
155		ovolssnul 25476	. . . . . . . . . 10 ⊢ (({𝑥 ∈ 𝐴 ∣ (¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝑏)} ⊆ (𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹})) ∧ (𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹})) ⊆ ℝ ∧ (vol*‘(𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹}))) = 0) → (vol*‘{𝑥 ∈ 𝐴 ∣ (¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝑏)}) = 0)
156	155	3expa 1125	. . . . . . . . 9 ⊢ ((({𝑥 ∈ 𝐴 ∣ (¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝑏)} ⊆ (𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹})) ∧ (𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹})) ⊆ ℝ) ∧ (vol*‘(𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹}))) = 0) → (vol*‘{𝑥 ∈ 𝐴 ∣ (¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝑏)}) = 0)
157	154, 156	sylan 587	. . . . . . . 8 ⊢ ((𝐴 ∈ dom vol ∧ (vol*‘(𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹}))) = 0) → (vol*‘{𝑥 ∈ 𝐴 ∣ (¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝑏)}) = 0)
158		nulmbl 25524	. . . . . . . 8 ⊢ (({𝑥 ∈ 𝐴 ∣ (¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝑏)} ⊆ ℝ ∧ (vol*‘{𝑥 ∈ 𝐴 ∣ (¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝑏)}) = 0) → {𝑥 ∈ 𝐴 ∣ (¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝑏)} ∈ dom vol)
159	142, 157, 158	syl2an2r 692	. . . . . . 7 ⊢ ((𝐴 ∈ dom vol ∧ (vol*‘(𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹}))) = 0) → {𝑥 ∈ 𝐴 ∣ (¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝑏)} ∈ dom vol)
160		unmbl 25526	. . . . . . 7 ⊢ (({𝑥 ∈ 𝐴 ∣ ∃𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)(𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))} ∈ dom vol ∧ {𝑥 ∈ 𝐴 ∣ (¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝑏)} ∈ dom vol) → ({𝑥 ∈ 𝐴 ∣ ∃𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)(𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))} ∪ {𝑥 ∈ 𝐴 ∣ (¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝑏)}) ∈ dom vol)
161	140, 159, 160	syl2anc 591	. . . . . 6 ⊢ ((𝐴 ∈ dom vol ∧ (vol*‘(𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹}))) = 0) → ({𝑥 ∈ 𝐴 ∣ ∃𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)(𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))} ∪ {𝑥 ∈ 𝐴 ∣ (¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝑏)}) ∈ dom vol)
162	161	3adant1 1137	. . . . 5 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol ∧ (vol*‘(𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹}))) = 0) → ({𝑥 ∈ 𝐴 ∣ ∃𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)(𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))} ∪ {𝑥 ∈ 𝐴 ∣ (¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝑏)}) ∈ dom vol)
163	162	adantr 482	. . . 4 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol ∧ (vol*‘(𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹}))) = 0) ∧ 𝑏 ∈ ran (,)) → ({𝑥 ∈ 𝐴 ∣ ∃𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)(𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))} ∪ {𝑥 ∈ 𝐴 ∣ (¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝑏)}) ∈ dom vol)
164	45, 163	eqeltrd 2841	. . 3 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol ∧ (vol*‘(𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹}))) = 0) ∧ 𝑏 ∈ ran (,)) → (◡𝐹 “ 𝑏) ∈ dom vol)
165	164	ralrimiva 3133	. 2 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol ∧ (vol*‘(𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹}))) = 0) → ∀𝑏 ∈ ran (,)(◡𝐹 “ 𝑏) ∈ dom vol)
166		ismbf 25617	. . 3 ⊢ (𝐹:𝐴⟶ℝ → (𝐹 ∈ MblFn ↔ ∀𝑏 ∈ ran (,)(◡𝐹 “ 𝑏) ∈ dom vol))
167	166	3ad2ant1 1140	. 2 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol ∧ (vol*‘(𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹}))) = 0) → (𝐹 ∈ MblFn ↔ ∀𝑏 ∈ ran (,)(◡𝐹 “ 𝑏) ∈ dom vol))
168	165, 167	mpbird 259	1 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol ∧ (vol*‘(𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹}))) = 0) → 𝐹 ∈ MblFn)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 397   ∨ wo 854   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121  {cab 2719  ∀wral 3055  ∃wrex 3065  {crab 3393   ∖ cdif 3882   ∪ cun 3883   ∩ cin 3884   ⊆ wss 3885  ∅c0 4264  ifcif 4457  𝒫 cpw 4532  {csn 4558  ∪ ciun 4924   class class class wbr 5075   ↦ cmpt 5156   E cep 5520  ^◡ccnv 5620  dom cdm 5621  ran crn 5622   “ cima 5624   ∘ ccom 5625  Rel wrel 5626   Fn wfn 6484  ⟶wf 6485  ‘cfv 6489  (class class class)co 7360   ↑_m cmap 8767  ℝcr 11032  0cc0 11033  (,)cioo 13293   ↾_t crest 17378  topGenctg 17395  Topctop 22880  TopOnctopon 22897  TopBasesctb 22932   CnP ccnp 23212  vol*covol 25451  volcvol 25452  MblFncmbf 25603

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682  ax-inf2 9557  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110  ax-pre-sup 11111

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-int 4881  df-iun 4926  df-disj 5043  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-se 5575  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-isom 6498  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-of 7624  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-oadd 8403  df-omul 8404  df-er 8637  df-map 8769  df-pm 8770  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-fi 9318  df-sup 9349  df-inf 9350  df-oi 9419  df-dju 9820  df-card 9858  df-acn 9861  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-div 11803  df-nn 12170  df-2 12239  df-3 12240  df-4 12241  df-n0 12433  df-z 12520  df-uz 12784  df-q 12894  df-rp 12938  df-xneg 13058  df-xadd 13059  df-xmul 13060  df-ioo 13297  df-ico 13299  df-icc 13300  df-fz 13457  df-fzo 13604  df-fl 13746  df-seq 13959  df-exp 14019  df-hash 14288  df-cj 15056  df-re 15057  df-im 15058  df-sqrt 15192  df-abs 15193  df-clim 15445  df-rlim 15446  df-sum 15644  df-rest 17380  df-topgen 17401  df-psmet 21343  df-xmet 21344  df-met 21345  df-bl 21346  df-mopn 21347  df-top 22881  df-topon 22898  df-bases 22933  df-cnp 23215  df-cmp 23374  df-ovol 25453  df-vol 25454  df-mbf 25608

This theorem is referenced by: (None)