Theorem cnambfre 37628

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 6990	. . . . . . . . 9 ⊢ (𝐹:𝐴⟶ℝ → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
3	2	cnveqd 5900	. . . . . . . 8 ⊢ (𝐹:𝐴⟶ℝ → ◡𝐹 = ◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
4	3	imaeq1d 6088	. . . . . . 7 ⊢ (𝐹:𝐴⟶ℝ → (◡𝐹 “ 𝑏) = (◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) “ 𝑏))
5	4	ad2antrr 725	. . . . . 6 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol) ∧ 𝑏 ∈ ran (,)) → (◡𝐹 “ 𝑏) = (◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) “ 𝑏))
6		exmid 893	. . . . . . . . . . 11 ⊢ (𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∨ ¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥))
7	6	biantrur 530	. . . . . . . . . 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 1009	. . . . . . . . . 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 275	. . . . . . . . 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 24802	. . . . . . . . . . . . . . . . . 18 ⊢ ran (,) ∈ TopBases
11		bastg 22994	. . . . . . . . . . . . . . . . . 18 ⊢ (ran (,) ∈ TopBases → ran (,) ⊆ (topGen‘ran (,)))
12	10, 11	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ran (,) ⊆ (topGen‘ran (,))
13	12	sseli 4004	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ∈ ran (,) → 𝑏 ∈ (topGen‘ran (,)))
14	13	ad2antlr 726	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol) ∧ 𝑏 ∈ ran (,)) ∧ 𝑥 ∈ 𝐴) → 𝑏 ∈ (topGen‘ran (,)))
15		cnpimaex 23285	. . . . . . . . . . . . . . . . 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 1123	. . . . . . . . . . . . . . . 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 1118	. . . . . . . . . . . . . . 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 679	. . . . . . . . . . . . . 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 412	. . . . . . . . . . . . 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 781	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol) ∧ (𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))) → (𝐹 “ 𝑦) ⊆ 𝑏)
21		ffn 6747	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹:𝐴⟶ℝ → 𝐹 Fn 𝐴)
22	21	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol) → 𝐹 Fn 𝐴)
23		restsspw 17491	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((topGen‘ran (,)) ↾t 𝐴) ⊆ 𝒫 𝐴
24	23	sseli 4004	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) → 𝑦 ∈ 𝒫 𝐴)
25	24	elpwid 4631	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) → 𝑦 ⊆ 𝐴)
26		simpl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏) → 𝑥 ∈ 𝑦)
27		fnfvima 7270	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑦 ⊆ 𝐴 ∧ 𝑥 ∈ 𝑦) → (𝐹‘𝑥) ∈ (𝐹 “ 𝑦))
28	22, 25, 26, 27	syl3an 1160	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol) ∧ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)) → (𝐹‘𝑥) ∈ (𝐹 “ 𝑦))
29	28	3expb 1120	. . . . . . . . . . . . . . . 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 4009	. . . . . . . . . . . . . . 15 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol) ∧ (𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))) → (𝐹‘𝑥) ∈ 𝑏)
31	30	rexlimdvaa 3162	. . . . . . . . . . . . . 14 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol) → (∃𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)(𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏) → (𝐹‘𝑥) ∈ 𝑏))
32	31	ad3antrrr 729	. . . . . . . . . . . . 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 212	. . . . . . . . . . . 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 578	. . . . . . . . . . 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 3197	. . . . . . . . . . 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 289	. . . . . . . . . 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 915	. . . . . . . . 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 283	. . . . . . . 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 3450	. . . . . . 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 2740	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))
41	40	mptpreima 6269	. . . . . . 7 ⊢ (◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) “ 𝑏) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ∈ 𝑏}
42		unrab 4334	. . . . . . 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 2805	. . . . . 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 2780	. . . . 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 1168	. . . 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 4230	. . . . . . . . 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 4297	. . . . . . . . 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 4335	. . . . . . . . . . . 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 5036	. . . . . . . . . 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 5094	. . . . . . . . . 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 5075	. . . . . . . . . 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 2776	. . . . . . . . 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 2776	. . . . . . . 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 2752	. . . . . . . . . . . 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 2752	. . . . . . . . . . . 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 780	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝐹 “ 𝑦) ⊆ 𝑏) ∧ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))) → 𝑥 ∈ 𝑦)
58	25	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝐹 “ 𝑦) ⊆ 𝑏) → 𝑦 ⊆ 𝐴)
59	58	sselda 4008	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝐹 “ 𝑦) ⊆ 𝑏) ∧ 𝑥 ∈ 𝑦) → 𝑥 ∈ 𝐴)
60		pm3.22 459	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹 “ 𝑦) ⊆ 𝑏 ∧ 𝑥 ∈ 𝑦) → (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))
61	60	adantll 713	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝐹 “ 𝑦) ⊆ 𝑏) ∧ 𝑥 ∈ 𝑦) → (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))
62	59, 61	jca 511	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝐹 “ 𝑦) ⊆ 𝑏) ∧ 𝑥 ∈ 𝑦) → (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)))
63	57, 62	impbida 800	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝐹 “ 𝑦) ⊆ 𝑏) → ((𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)) ↔ 𝑥 ∈ 𝑦))
64	63	abbidv 2811	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝐹 “ 𝑦) ⊆ 𝑏) → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))} = {𝑥 ∣ 𝑥 ∈ 𝑦})
65		df-rab 3444	. . . . . . . . . . . . 13 ⊢ {𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))}
66		cvjust 2734	. . . . . . . . . . . . 13 ⊢ 𝑦 = {𝑥 ∣ 𝑥 ∈ 𝑦}
67	64, 65, 66	3eqtr4g 2805	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝐹 “ 𝑦) ⊆ 𝑏) → {𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} = 𝑦)
68		simpr 484	. . . . . . . . . . . . . . . 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 3156	. . . . . . . . . . . . . 14 ⊢ (¬ (𝐹 “ 𝑦) ⊆ 𝑏 → ∀𝑥 ∈ 𝐴 ¬ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))
71		rabeq0 4411	. . . . . . . . . . . . . 14 ⊢ ({𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))
72	70, 71	sylibr 234	. . . . . . . . . . . . 13 ⊢ (¬ (𝐹 “ 𝑦) ⊆ 𝑏 → {𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} = ∅)
73	72	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ ¬ (𝐹 “ 𝑦) ⊆ 𝑏) → {𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} = ∅)
74	55, 56, 67, 73	ifbothda 4586	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) → {𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} = if((𝐹 “ 𝑦) ⊆ 𝑏, 𝑦, ∅))
75	74	iuneq2i 5036	. . . . . . . . . 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 24803	. . . . . . . . . . . . 13 ⊢ (topGen‘ran (,)) ∈ Top
77		resttop 23189	. . . . . . . . . . . . 13 ⊢ (((topGen‘ran (,)) ∈ Top ∧ 𝐴 ∈ dom vol) → ((topGen‘ran (,)) ↾t 𝐴) ∈ Top)
78	76, 77	mpan 689	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom vol → ((topGen‘ran (,)) ↾t 𝐴) ∈ Top)
79		0opn 22931	. . . . . . . . . . . . . . 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 4593	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ ∅ ∈ ((topGen‘ran (,)) ↾t 𝐴)) → if((𝐹 “ 𝑦) ⊆ 𝑏, 𝑦, ∅) ∈ ((topGen‘ran (,)) ↾t 𝐴))
82	81	ancoms 458	. . . . . . . . . . . . . 14 ⊢ ((∅ ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)) → if((𝐹 “ 𝑦) ⊆ 𝑏, 𝑦, ∅) ∈ ((topGen‘ran (,)) ↾t 𝐴))
83	80, 82	sylan 579	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ dom vol ∧ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)) → if((𝐹 “ 𝑦) ⊆ 𝑏, 𝑦, ∅) ∈ ((topGen‘ran (,)) ↾t 𝐴))
84	83	ralrimiva 3152	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom vol → ∀𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)if((𝐹 “ 𝑦) ⊆ 𝑏, 𝑦, ∅) ∈ ((topGen‘ran (,)) ↾t 𝐴))
85		iunopn 22925	. . . . . . . . . . . 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 583	. . . . . . . . . . 11 ⊢ (𝐴 ∈ dom vol → ∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)if((𝐹 “ 𝑦) ⊆ 𝑏, 𝑦, ∅) ∈ ((topGen‘ran (,)) ↾t 𝐴))
87		eqid 2740	. . . . . . . . . . . 12 ⊢ ((topGen‘ran (,)) ↾t 𝐴) = ((topGen‘ran (,)) ↾t 𝐴)
88	87	subopnmbl 25658	. . . . . . . . . . 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 686	. . . . . . . . . 10 ⊢ (𝐴 ∈ dom vol → ∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)if((𝐹 “ 𝑦) ⊆ 𝑏, 𝑦, ∅) ∈ dom vol)
90	75, 89	eqeltrid 2848	. . . . . . . . 9 ⊢ (𝐴 ∈ dom vol → ∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ∈ dom vol)
91		difss 4159	. . . . . . . . . . . 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 4103	. . . . . . . . . . . . . 14 ⊢ {𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ⊆ 𝐴
93	92	rgenw 3071	. . . . . . . . . . . . 13 ⊢ ∀𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ⊆ 𝐴
94		iunss 5068	. . . . . . . . . . . . 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 231	. . . . . . . . . . . 12 ⊢ ∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ⊆ 𝐴
96	91, 95	sstri 4018	. . . . . . . . . . 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 25585	. . . . . . . . . . 11 ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ)
98	96, 97	sstrid 4020	. . . . . . . . . 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 4167	. . . . . . . . . . . . . 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 5851	. . . . . . . . . . . . . . . . . . . 20 ⊢ Rel E
102		elrelimasn 6115	. . . . . . . . . . . . . . . . . . . 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 6933	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∈ V
105	104	epeli 5601	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹 E ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ↔ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥))
106	103, 105	bitr2i 276	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ↔ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∈ ( E “ {𝐹}))
107	106	anbi2i 622	. . . . . . . . . . . . . . . . 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 7481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℝ ↑m 𝐴) ∈ V
109	108	rabex 5357	. . . . . . . . . . . . . . . . . . . 20 ⊢ {𝑓 ∈ (ℝ ↑m 𝐴) ∣ ∀𝑏 ∈ (topGen‘ran (,))((𝑓‘𝑥) ∈ 𝑏 → ∃𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)(𝑥 ∈ 𝑦 ∧ (𝑓 “ 𝑦) ⊆ 𝑏))} ∈ V
110		eqid 2740	. . . . . . . . . . . . . . . . . . . 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 6723	. . . . . . . . . . . . . . . . . . 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 24805	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
113		resttopon 23190	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ 𝐴 ⊆ ℝ) → ((topGen‘ran (,)) ↾t 𝐴) ∈ (TopOn‘𝐴))
114	112, 97, 113	sylancr 586	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ∈ dom vol → ((topGen‘ran (,)) ↾t 𝐴) ∈ (TopOn‘𝐴))
115		cnpfval 23263	. . . . . . . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . . . . . . . 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 6672	. . . . . . . . . . . . . . . . . . 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 258	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ dom vol → (((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) Fn 𝐴)
119		elpreima 7091	. . . . . . . . . . . . . . . . . 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 290	. . . . . . . . . . . . . . . 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 2811	. . . . . . . . . . . . . . 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 3444	. . . . . . . . . . . . . . 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 6282	. . . . . . . . . . . . . . . 16 ⊢ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹}) = (◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) “ ( E “ {𝐹}))
125		abid2 2882	. . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . 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 2805	. . . . . . . . . . . . . 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 4149	. . . . . . . . . . . . 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 4061	. . . . . . . . . . . 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 4159	. . . . . . . . . . . . 13 ⊢ (𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹})) ⊆ 𝐴
131	130, 97	sstrid 4020	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom vol → (𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹})) ⊆ ℝ)
132	129, 131	jca 511	. . . . . . . . . . 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 25541	. . . . . . . . . . . 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 1118	. . . . . . . . . . 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 579	. . . . . . . . . 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 25589	. . . . . . . . . 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 684	. . . . . . . . 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 25597	. . . . . . . . 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 684	. . . . . . . 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 2849	. . . . . . 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 4103	. . . . . . . . 9 ⊢ {𝑥 ∈ 𝐴 ∣ (¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝑏)} ⊆ 𝐴
142	141, 97	sstrid 4020	. . . . . . . 8 ⊢ (𝐴 ∈ dom vol → {𝑥 ∈ 𝐴 ∣ (¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝑏)} ⊆ ℝ)
143	124	eleq2i 2836	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹}) ↔ 𝑥 ∈ (◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) “ ( E “ {𝐹})))
144		ibar 528	. . . . . . . . . . . . . . . . . 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 284	. . . . . . . . . . . . . . . . 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 509	. . . . . . . . . . . . . . . 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 284	. . . . . . . . . . . . . . 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 318	. . . . . . . . . . . . . 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 229	. . . . . . . . . . . . 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 491	. . . . . . . . . . . 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 4099	. . . . . . . . . . 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 3985	. . . . . . . . . . 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 4060	. . . . . . . . . 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 511	. . . . . . . . 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 25541	. . . . . . . . . 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 1118	. . . . . . . . 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 579	. . . . . . . 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 25589	. . . . . . . 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 684	. . . . . . 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 25591	. . . . . . 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 583	. . . . . 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 1130	. . . . 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 480	. . . 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 2844	. . 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 3152	. 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 25682	. . 3 ⊢ (𝐹:𝐴⟶ℝ → (𝐹 ∈ MblFn ↔ ∀𝑏 ∈ ran (,)(◡𝐹 “ 𝑏) ∈ dom vol))
167	166	3ad2ant1 1133	. 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 257	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 206   ∧ wa 395   ∨ wo 846   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  {cab 2717  ∀wral 3067  ∃wrex 3076  {crab 3443   ∖ cdif 3973   ∪ cun 3974   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  ifcif 4548  𝒫 cpw 4622  {csn 4648  ∪ ciun 5015   class class class wbr 5166   ↦ cmpt 5249   E cep 5598  ^◡ccnv 5699  dom cdm 5700  ran crn 5701   “ cima 5703   ∘ ccom 5704  Rel wrel 5705   Fn wfn 6568  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448   ↑_m cmap 8884  ℝcr 11183  0cc0 11184  (,)cioo 13407   ↾_t crest 17480  topGenctg 17497  Topctop 22920  TopOnctopon 22937  TopBasesctb 22973   CnP ccnp 23254  vol*covol 25516  volcvol 25517  MblFncmbf 25668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-disj 5134  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-of 7714  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-oadd 8526  df-omul 8527  df-er 8763  df-map 8886  df-pm 8887  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-fi 9480  df-sup 9511  df-inf 9512  df-oi 9579  df-dju 9970  df-card 10008  df-acn 10011  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-n0 12554  df-z 12640  df-uz 12904  df-q 13014  df-rp 13058  df-xneg 13175  df-xadd 13176  df-xmul 13177  df-ioo 13411  df-ico 13413  df-icc 13414  df-fz 13568  df-fzo 13712  df-fl 13843  df-seq 14053  df-exp 14113  df-hash 14380  df-cj 15148  df-re 15149  df-im 15150  df-sqrt 15284  df-abs 15285  df-clim 15534  df-rlim 15535  df-sum 15735  df-rest 17482  df-topgen 17503  df-psmet 21379  df-xmet 21380  df-met 21381  df-bl 21382  df-mopn 21383  df-top 22921  df-topon 22938  df-bases 22974  df-cnp 23257  df-cmp 23416  df-ovol 25518  df-vol 25519  df-mbf 25673

This theorem is referenced by: (None)