Theorem cnambfre 37369

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 6971	. . . . . . . . 9 ⊢ (𝐹:𝐴⟶ℝ → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
3	2	cnveqd 5882	. . . . . . . 8 ⊢ (𝐹:𝐴⟶ℝ → ◡𝐹 = ◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
4	3	imaeq1d 6068	. . . . . . 7 ⊢ (𝐹:𝐴⟶ℝ → (◡𝐹 “ 𝑏) = (◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) “ 𝑏))
5	4	ad2antrr 724	. . . . . 6 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol) ∧ 𝑏 ∈ ran (,)) → (◡𝐹 “ 𝑏) = (◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) “ 𝑏))
6		exmid 892	. . . . . . . . . . 11 ⊢ (𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∨ ¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥))
7	6	biantrur 529	. . . . . . . . . 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 1006	. . . . . . . . . 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 274	. . . . . . . . 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 24768	. . . . . . . . . . . . . . . . . 18 ⊢ ran (,) ∈ TopBases
11		bastg 22960	. . . . . . . . . . . . . . . . . 18 ⊢ (ran (,) ∈ TopBases → ran (,) ⊆ (topGen‘ran (,)))
12	10, 11	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ran (,) ⊆ (topGen‘ran (,))
13	12	sseli 3975	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ∈ ran (,) → 𝑏 ∈ (topGen‘ran (,)))
14	13	ad2antlr 725	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol) ∧ 𝑏 ∈ ran (,)) ∧ 𝑥 ∈ 𝐴) → 𝑏 ∈ (topGen‘ran (,)))
15		cnpimaex 23251	. . . . . . . . . . . . . . . . 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 1120	. . . . . . . . . . . . . . . 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 1115	. . . . . . . . . . . . . . 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 678	. . . . . . . . . . . . . 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 411	. . . . . . . . . . . . 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 780	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol) ∧ (𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))) → (𝐹 “ 𝑦) ⊆ 𝑏)
21		ffn 6728	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹:𝐴⟶ℝ → 𝐹 Fn 𝐴)
22	21	adantr 479	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol) → 𝐹 Fn 𝐴)
23		restsspw 17446	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((topGen‘ran (,)) ↾t 𝐴) ⊆ 𝒫 𝐴
24	23	sseli 3975	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) → 𝑦 ∈ 𝒫 𝐴)
25	24	elpwid 4616	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) → 𝑦 ⊆ 𝐴)
26		simpl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏) → 𝑥 ∈ 𝑦)
27		fnfvima 7250	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑦 ⊆ 𝐴 ∧ 𝑥 ∈ 𝑦) → (𝐹‘𝑥) ∈ (𝐹 “ 𝑦))
28	22, 25, 26, 27	syl3an 1157	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol) ∧ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)) → (𝐹‘𝑥) ∈ (𝐹 “ 𝑦))
29	28	3expb 1117	. . . . . . . . . . . . . . . 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 3980	. . . . . . . . . . . . . . 15 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol) ∧ (𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))) → (𝐹‘𝑥) ∈ 𝑏)
31	30	rexlimdvaa 3146	. . . . . . . . . . . . . 14 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol) → (∃𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)(𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏) → (𝐹‘𝑥) ∈ 𝑏))
32	31	ad3antrrr 728	. . . . . . . . . . . . 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 211	. . . . . . . . . . . 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 577	. . . . . . . . . . 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 3181	. . . . . . . . . . 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 288	. . . . . . . . . 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 914	. . . . . . . . 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 282	. . . . . . . 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 3426	. . . . . . 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 2726	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))
41	40	mptpreima 6249	. . . . . . 7 ⊢ (◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) “ 𝑏) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ∈ 𝑏}
42		unrab 4307	. . . . . . 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 2791	. . . . . 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 2766	. . . . 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 1165	. . . 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 4202	. . . . . . . . 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 4269	. . . . . . . . 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 4308	. . . . . . . . . . . 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 5022	. . . . . . . . . 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 5079	. . . . . . . . . 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 5060	. . . . . . . . . 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 2762	. . . . . . . . 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 2762	. . . . . . . 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 2738	. . . . . . . . . . . 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 2738	. . . . . . . . . . . 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 779	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝐹 “ 𝑦) ⊆ 𝑏) ∧ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))) → 𝑥 ∈ 𝑦)
58	25	adantr 479	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝐹 “ 𝑦) ⊆ 𝑏) → 𝑦 ⊆ 𝐴)
59	58	sselda 3979	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝐹 “ 𝑦) ⊆ 𝑏) ∧ 𝑥 ∈ 𝑦) → 𝑥 ∈ 𝐴)
60		pm3.22 458	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹 “ 𝑦) ⊆ 𝑏 ∧ 𝑥 ∈ 𝑦) → (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))
61	60	adantll 712	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝐹 “ 𝑦) ⊆ 𝑏) ∧ 𝑥 ∈ 𝑦) → (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))
62	59, 61	jca 510	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝐹 “ 𝑦) ⊆ 𝑏) ∧ 𝑥 ∈ 𝑦) → (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)))
63	57, 62	impbida 799	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝐹 “ 𝑦) ⊆ 𝑏) → ((𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)) ↔ 𝑥 ∈ 𝑦))
64	63	abbidv 2795	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝐹 “ 𝑦) ⊆ 𝑏) → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))} = {𝑥 ∣ 𝑥 ∈ 𝑦})
65		df-rab 3420	. . . . . . . . . . . . 13 ⊢ {𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))}
66		cvjust 2720	. . . . . . . . . . . . 13 ⊢ 𝑦 = {𝑥 ∣ 𝑥 ∈ 𝑦}
67	64, 65, 66	3eqtr4g 2791	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝐹 “ 𝑦) ⊆ 𝑏) → {𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} = 𝑦)
68		simpr 483	. . . . . . . . . . . . . . . 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 3140	. . . . . . . . . . . . . 14 ⊢ (¬ (𝐹 “ 𝑦) ⊆ 𝑏 → ∀𝑥 ∈ 𝐴 ¬ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))
71		rabeq0 4389	. . . . . . . . . . . . . 14 ⊢ ({𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))
72	70, 71	sylibr 233	. . . . . . . . . . . . 13 ⊢ (¬ (𝐹 “ 𝑦) ⊆ 𝑏 → {𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} = ∅)
73	72	adantl 480	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ ¬ (𝐹 “ 𝑦) ⊆ 𝑏) → {𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} = ∅)
74	55, 56, 67, 73	ifbothda 4571	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) → {𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} = if((𝐹 “ 𝑦) ⊆ 𝑏, 𝑦, ∅))
75	74	iuneq2i 5022	. . . . . . . . . 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 24769	. . . . . . . . . . . . 13 ⊢ (topGen‘ran (,)) ∈ Top
77		resttop 23155	. . . . . . . . . . . . 13 ⊢ (((topGen‘ran (,)) ∈ Top ∧ 𝐴 ∈ dom vol) → ((topGen‘ran (,)) ↾t 𝐴) ∈ Top)
78	76, 77	mpan 688	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom vol → ((topGen‘ran (,)) ↾t 𝐴) ∈ Top)
79		0opn 22897	. . . . . . . . . . . . . . 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 4578	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ ∅ ∈ ((topGen‘ran (,)) ↾t 𝐴)) → if((𝐹 “ 𝑦) ⊆ 𝑏, 𝑦, ∅) ∈ ((topGen‘ran (,)) ↾t 𝐴))
82	81	ancoms 457	. . . . . . . . . . . . . 14 ⊢ ((∅ ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)) → if((𝐹 “ 𝑦) ⊆ 𝑏, 𝑦, ∅) ∈ ((topGen‘ran (,)) ↾t 𝐴))
83	80, 82	sylan 578	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ dom vol ∧ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)) → if((𝐹 “ 𝑦) ⊆ 𝑏, 𝑦, ∅) ∈ ((topGen‘ran (,)) ↾t 𝐴))
84	83	ralrimiva 3136	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom vol → ∀𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)if((𝐹 “ 𝑦) ⊆ 𝑏, 𝑦, ∅) ∈ ((topGen‘ran (,)) ↾t 𝐴))
85		iunopn 22891	. . . . . . . . . . . 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 582	. . . . . . . . . . 11 ⊢ (𝐴 ∈ dom vol → ∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)if((𝐹 “ 𝑦) ⊆ 𝑏, 𝑦, ∅) ∈ ((topGen‘ran (,)) ↾t 𝐴))
87		eqid 2726	. . . . . . . . . . . 12 ⊢ ((topGen‘ran (,)) ↾t 𝐴) = ((topGen‘ran (,)) ↾t 𝐴)
88	87	subopnmbl 25624	. . . . . . . . . . 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 685	. . . . . . . . . 10 ⊢ (𝐴 ∈ dom vol → ∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)if((𝐹 “ 𝑦) ⊆ 𝑏, 𝑦, ∅) ∈ dom vol)
90	75, 89	eqeltrid 2830	. . . . . . . . 9 ⊢ (𝐴 ∈ dom vol → ∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ∈ dom vol)
91		difss 4131	. . . . . . . . . . . 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 4076	. . . . . . . . . . . . . 14 ⊢ {𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ⊆ 𝐴
93	92	rgenw 3055	. . . . . . . . . . . . 13 ⊢ ∀𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ⊆ 𝐴
94		iunss 5053	. . . . . . . . . . . . 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 230	. . . . . . . . . . . 12 ⊢ ∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ⊆ 𝐴
96	91, 95	sstri 3989	. . . . . . . . . . 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 25551	. . . . . . . . . . 11 ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ)
98	96, 97	sstrid 3991	. . . . . . . . . 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 4139	. . . . . . . . . . . . . 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 5833	. . . . . . . . . . . . . . . . . . . 20 ⊢ Rel E
102		elrelimasn 6095	. . . . . . . . . . . . . . . . . . . 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 6914	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∈ V
105	104	epeli 5588	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹 E ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ↔ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥))
106	103, 105	bitr2i 275	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ↔ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∈ ( E “ {𝐹}))
107	106	anbi2i 621	. . . . . . . . . . . . . . . . 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 7457	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℝ ↑m 𝐴) ∈ V
109	108	rabex 5339	. . . . . . . . . . . . . . . . . . . 20 ⊢ {𝑓 ∈ (ℝ ↑m 𝐴) ∣ ∀𝑏 ∈ (topGen‘ran (,))((𝑓‘𝑥) ∈ 𝑏 → ∃𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)(𝑥 ∈ 𝑦 ∧ (𝑓 “ 𝑦) ⊆ 𝑏))} ∈ V
110		eqid 2726	. . . . . . . . . . . . . . . . . . . 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 6704	. . . . . . . . . . . . . . . . . . 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 24771	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
113		resttopon 23156	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ 𝐴 ⊆ ℝ) → ((topGen‘ran (,)) ↾t 𝐴) ∈ (TopOn‘𝐴))
114	112, 97, 113	sylancr 585	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ∈ dom vol → ((topGen‘ran (,)) ↾t 𝐴) ∈ (TopOn‘𝐴))
115		cnpfval 23229	. . . . . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . . . . 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 6653	. . . . . . . . . . . . . . . . . . 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 257	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ dom vol → (((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) Fn 𝐴)
119		elpreima 7071	. . . . . . . . . . . . . . . . . 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 289	. . . . . . . . . . . . . . . 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 2795	. . . . . . . . . . . . . . 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 3420	. . . . . . . . . . . . . . 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 6262	. . . . . . . . . . . . . . . 16 ⊢ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹}) = (◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) “ ( E “ {𝐹}))
125		abid2 2864	. . . . . . . . . . . . . . . 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 2757	. . . . . . . . . . . . . . 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 2791	. . . . . . . . . . . . . 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 4121	. . . . . . . . . . . . 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 4032	. . . . . . . . . . . 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 4131	. . . . . . . . . . . . 13 ⊢ (𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹})) ⊆ 𝐴
131	130, 97	sstrid 3991	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom vol → (𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹})) ⊆ ℝ)
132	129, 131	jca 510	. . . . . . . . . . 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 25507	. . . . . . . . . . . 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 1115	. . . . . . . . . . 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 578	. . . . . . . . . 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 25555	. . . . . . . . . 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 683	. . . . . . . . 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 25563	. . . . . . . . 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 683	. . . . . . . 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 2831	. . . . . . 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 4076	. . . . . . . . 9 ⊢ {𝑥 ∈ 𝐴 ∣ (¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝑏)} ⊆ 𝐴
142	141, 97	sstrid 3991	. . . . . . . 8 ⊢ (𝐴 ∈ dom vol → {𝑥 ∈ 𝐴 ∣ (¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝑏)} ⊆ ℝ)
143	124	eleq2i 2818	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹}) ↔ 𝑥 ∈ (◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) “ ( E “ {𝐹})))
144		ibar 527	. . . . . . . . . . . . . . . . . 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 283	. . . . . . . . . . . . . . . . 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 508	. . . . . . . . . . . . . . . 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 283	. . . . . . . . . . . . . . 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 317	. . . . . . . . . . . . . 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 228	. . . . . . . . . . . . 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 490	. . . . . . . . . . . 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 4072	. . . . . . . . . . 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 3956	. . . . . . . . . . 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 4031	. . . . . . . . . 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 510	. . . . . . . . 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 25507	. . . . . . . . . 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 1115	. . . . . . . . 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 578	. . . . . . . 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 25555	. . . . . . . 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 683	. . . . . . 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 25557	. . . . . . 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 582	. . . . . 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 1127	. . . . 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 479	. . . 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 2826	. . 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 3136	. 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 25648	. . 3 ⊢ (𝐹:𝐴⟶ℝ → (𝐹 ∈ MblFn ↔ ∀𝑏 ∈ ran (,)(◡𝐹 “ 𝑏) ∈ dom vol))
167	166	3ad2ant1 1130	. 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 256	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 205   ∧ wa 394   ∨ wo 845   ∧ w3a 1084   = wceq 1534   ∈ wcel 2099  {cab 2703  ∀wral 3051  ∃wrex 3060  {crab 3419   ∖ cdif 3944   ∪ cun 3945   ∩ cin 3946   ⊆ wss 3947  ∅c0 4325  ifcif 4533  𝒫 cpw 4607  {csn 4633  ∪ ciun 5001   class class class wbr 5153   ↦ cmpt 5236   E cep 5585  ^◡ccnv 5681  dom cdm 5682  ran crn 5683   “ cima 5685   ∘ ccom 5686  Rel wrel 5687   Fn wfn 6549  ⟶wf 6550  ‘cfv 6554  (class class class)co 7424   ↑_m cmap 8855  ℝcr 11157  0cc0 11158  (,)cioo 13378   ↾_t crest 17435  topGenctg 17452  Topctop 22886  TopOnctopon 22903  TopBasesctb 22939   CnP ccnp 23220  vol*covol 25482  volcvol 25483  MblFncmbf 25634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-inf2 9684  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235  ax-pre-sup 11236

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-disj 5119  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-isom 6563  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-of 7690  df-om 7877  df-1st 8003  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-2o 8497  df-oadd 8500  df-omul 8501  df-er 8734  df-map 8857  df-pm 8858  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-fi 9454  df-sup 9485  df-inf 9486  df-oi 9553  df-dju 9944  df-card 9982  df-acn 9985  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-div 11922  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-n0 12525  df-z 12611  df-uz 12875  df-q 12985  df-rp 13029  df-xneg 13146  df-xadd 13147  df-xmul 13148  df-ioo 13382  df-ico 13384  df-icc 13385  df-fz 13539  df-fzo 13682  df-fl 13812  df-seq 14022  df-exp 14082  df-hash 14348  df-cj 15104  df-re 15105  df-im 15106  df-sqrt 15240  df-abs 15241  df-clim 15490  df-rlim 15491  df-sum 15691  df-rest 17437  df-topgen 17458  df-psmet 21335  df-xmet 21336  df-met 21337  df-bl 21338  df-mopn 21339  df-top 22887  df-topon 22904  df-bases 22940  df-cnp 23223  df-cmp 23382  df-ovol 25484  df-vol 25485  df-mbf 25639

This theorem is referenced by: (None)