Theorem cnambfre 34944

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 6736	. . . . . . . . 9 ⊢ (𝐹:𝐴⟶ℝ → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
3	2	cnveqd 5749	. . . . . . . 8 ⊢ (𝐹:𝐴⟶ℝ → ◡𝐹 = ◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
4	3	imaeq1d 5931	. . . . . . 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 891	. . . . . . . . . . 11 ⊢ (𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∨ ¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥))
7	6	biantrur 533	. . . . . . . . . 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 1005	. . . . . . . . . 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 23372	. . . . . . . . . . . . . . . . . 18 ⊢ ran (,) ∈ TopBases
11		bastg 21577	. . . . . . . . . . . . . . . . . 18 ⊢ (ran (,) ∈ TopBases → ran (,) ⊆ (topGen‘ran (,)))
12	10, 11	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ran (,) ⊆ (topGen‘ran (,))
13	12	sseli 3966	. . . . . . . . . . . . . . . 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 21867	. . . . . . . . . . . . . . . . 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 1119	. . . . . . . . . . . . . . . 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 1114	. . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . 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 6517	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹:𝐴⟶ℝ → 𝐹 Fn 𝐴)
22	21	adantr 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol) → 𝐹 Fn 𝐴)
23		restsspw 16708	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((topGen‘ran (,)) ↾t 𝐴) ⊆ 𝒫 𝐴
24	23	sseli 3966	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) → 𝑦 ∈ 𝒫 𝐴)
25	24	elpwid 4553	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) → 𝑦 ⊆ 𝐴)
26		simpl 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏) → 𝑥 ∈ 𝑦)
27		fnfvima 6998	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑦 ⊆ 𝐴 ∧ 𝑥 ∈ 𝑦) → (𝐹‘𝑥) ∈ (𝐹 “ 𝑦))
28	22, 25, 26, 27	syl3an 1156	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol) ∧ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)) → (𝐹‘𝑥) ∈ (𝐹 “ 𝑦))
29	28	3expb 1116	. . . . . . . . . . . . . . . 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 3971	. . . . . . . . . . . . . . 15 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol) ∧ (𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))) → (𝐹‘𝑥) ∈ 𝑏)
31	30	rexlimdvaa 3288	. . . . . . . . . . . . . 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 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 581	. . . . . . . . . . 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 3353	. . . . . . . . . . 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	syl6bbr 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 913	. . . . . . . . 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	syl5bb 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 3481	. . . . . . 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 2824	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))
41	40	mptpreima 6095	. . . . . . 7 ⊢ (◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) “ 𝑏) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ∈ 𝑏}
42		unrab 4277	. . . . . . 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 2884	. . . . . 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 2859	. . . . 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 1164	. . . 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 4181	. . . . . . . . 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 4247	. . . . . . . . 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 4278	. . . . . . . . . . . 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 4943	. . . . . . . . . 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 4996	. . . . . . . . . 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 4979	. . . . . . . . . 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 2855	. . . . . . . . 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 2855	. . . . . . . 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 2836	. . . . . . . . . . . 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 2836	. . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝐹 “ 𝑦) ⊆ 𝑏) → 𝑦 ⊆ 𝐴)
59	58	sselda 3970	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝐹 “ 𝑦) ⊆ 𝑏) ∧ 𝑥 ∈ 𝑦) → 𝑥 ∈ 𝐴)
60		pm3.22 462	. . . . . . . . . . . . . . . . 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 514	. . . . . . . . . . . . . . 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 2888	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝐹 “ 𝑦) ⊆ 𝑏) → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))} = {𝑥 ∣ 𝑥 ∈ 𝑦})
65		df-rab 3150	. . . . . . . . . . . . 13 ⊢ {𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))}
66		cvjust 2819	. . . . . . . . . . . . 13 ⊢ 𝑦 = {𝑥 ∣ 𝑥 ∈ 𝑦}
67	64, 65, 66	3eqtr4g 2884	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝐹 “ 𝑦) ⊆ 𝑏) → {𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} = 𝑦)
68		simpr 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏) → (𝐹 “ 𝑦) ⊆ 𝑏)
69	68	con3i 157	. . . . . . . . . . . . . . 15 ⊢ (¬ (𝐹 “ 𝑦) ⊆ 𝑏 → ¬ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))
70	69	ralrimivw 3186	. . . . . . . . . . . . . 14 ⊢ (¬ (𝐹 “ 𝑦) ⊆ 𝑏 → ∀𝑥 ∈ 𝐴 ¬ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))
71		rabeq0 4341	. . . . . . . . . . . . . 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 484	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ ¬ (𝐹 “ 𝑦) ⊆ 𝑏) → {𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} = ∅)
74	55, 56, 67, 73	ifbothda 4507	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) → {𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} = if((𝐹 “ 𝑦) ⊆ 𝑏, 𝑦, ∅))
75	74	iuneq2i 4943	. . . . . . . . . 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 23373	. . . . . . . . . . . . 13 ⊢ (topGen‘ran (,)) ∈ Top
77		resttop 21771	. . . . . . . . . . . . 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 21515	. . . . . . . . . . . . . . 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 4514	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ ∅ ∈ ((topGen‘ran (,)) ↾t 𝐴)) → if((𝐹 “ 𝑦) ⊆ 𝑏, 𝑦, ∅) ∈ ((topGen‘ran (,)) ↾t 𝐴))
82	81	ancoms 461	. . . . . . . . . . . . . 14 ⊢ ((∅ ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)) → if((𝐹 “ 𝑦) ⊆ 𝑏, 𝑦, ∅) ∈ ((topGen‘ran (,)) ↾t 𝐴))
83	80, 82	sylan 582	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ dom vol ∧ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)) → if((𝐹 “ 𝑦) ⊆ 𝑏, 𝑦, ∅) ∈ ((topGen‘ran (,)) ↾t 𝐴))
84	83	ralrimiva 3185	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom vol → ∀𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)if((𝐹 “ 𝑦) ⊆ 𝑏, 𝑦, ∅) ∈ ((topGen‘ran (,)) ↾t 𝐴))
85		iunopn 21509	. . . . . . . . . . . 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 586	. . . . . . . . . . 11 ⊢ (𝐴 ∈ dom vol → ∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)if((𝐹 “ 𝑦) ⊆ 𝑏, 𝑦, ∅) ∈ ((topGen‘ran (,)) ↾t 𝐴))
87		eqid 2824	. . . . . . . . . . . 12 ⊢ ((topGen‘ran (,)) ↾t 𝐴) = ((topGen‘ran (,)) ↾t 𝐴)
88	87	subopnmbl 24208	. . . . . . . . . . 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 2920	. . . . . . . . 9 ⊢ (𝐴 ∈ dom vol → ∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ∈ dom vol)
91		difss 4111	. . . . . . . . . . . 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 4059	. . . . . . . . . . . . . 14 ⊢ {𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ⊆ 𝐴
93	92	rgenw 3153	. . . . . . . . . . . . 13 ⊢ ∀𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ⊆ 𝐴
94		iunss 4972	. . . . . . . . . . . . 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 3979	. . . . . . . . . . 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 24135	. . . . . . . . . . 11 ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ)
98	96, 97	sstrid 3981	. . . . . . . . . 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 4119	. . . . . . . . . . . . . 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		ovex 7192	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℝ ↑m 𝐴) ∈ V
102	101	rabex 5238	. . . . . . . . . . . . . . . . . . . 20 ⊢ {𝑓 ∈ (ℝ ↑m 𝐴) ∣ ∀𝑏 ∈ (topGen‘ran (,))((𝑓‘𝑥) ∈ 𝑏 → ∃𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)(𝑥 ∈ 𝑦 ∧ (𝑓 “ 𝑦) ⊆ 𝑏))} ∈ V
103		eqid 2824	. . . . . . . . . . . . . . . . . . . 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))})
104	102, 103	fnmpti 6494	. . . . . . . . . . . . . . . . . . 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
105		retopon 23375	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
106		resttopon 21772	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ 𝐴 ⊆ ℝ) → ((topGen‘ran (,)) ↾t 𝐴) ∈ (TopOn‘𝐴))
107	105, 97, 106	sylancr 589	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ∈ dom vol → ((topGen‘ran (,)) ↾t 𝐴) ∈ (TopOn‘𝐴))
108		cnpfval 21845	. . . . . . . . . . . . . . . . . . . . 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))}))
109	107, 105, 108	sylancl 588	. . . . . . . . . . . . . . . . . . . 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))}))
110	109	fneq1d 6449	. . . . . . . . . . . . . . . . . . 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))
111	104, 110	mpbiri 260	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ dom vol → (((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) Fn 𝐴)
112		elpreima 6831	. . . . . . . . . . . . . . . . . 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}))))
113	111, 112	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}))))
114		rele 5702	. . . . . . . . . . . . . . . . . . . 20 ⊢ Rel E
115		elrelimasn 5956	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Rel E → (((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∈ ( E “ {𝐹}) ↔ 𝐹 E ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)))
116	114, 115	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∈ ( E “ {𝐹}) ↔ 𝐹 E ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥))
117		fvex 6686	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∈ V
118	117	epeli 5471	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹 E ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ↔ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥))
119	116, 118	bitr2i 278	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ↔ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∈ ( E “ {𝐹}))
120	119	anbi2i 624	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)) ↔ (𝑥 ∈ 𝐴 ∧ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∈ ( E “ {𝐹})))
121	113, 120	syl6rbbr 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 2888	. . . . . . . . . . . . . . 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 3150	. . . . . . . . . . . . . . 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 6107	. . . . . . . . . . . . . . . 16 ⊢ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹}) = (◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) “ ( E “ {𝐹}))
125		abid2 2960	. . . . . . . . . . . . . . . 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 2850	. . . . . . . . . . . . . . 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 2884	. . . . . . . . . . . . . 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 4102	. . . . . . . . . . . . 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 4022	. . . . . . . . . . . 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 4111	. . . . . . . . . . . . 13 ⊢ (𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹})) ⊆ 𝐴
131	130, 97	sstrid 3981	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom vol → (𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹})) ⊆ ℝ)
132	129, 131	jca 514	. . . . . . . . . . 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 24091	. . . . . . . . . . . 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 1114	. . . . . . . . . . 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 582	. . . . . . . . . 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 24139	. . . . . . . . . 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 24147	. . . . . . . . 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 2921	. . . . . . 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 4059	. . . . . . . . 9 ⊢ {𝑥 ∈ 𝐴 ∣ (¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝑏)} ⊆ 𝐴
142	141, 97	sstrid 3981	. . . . . . . 8 ⊢ (𝐴 ∈ dom vol → {𝑥 ∈ 𝐴 ∣ (¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝑏)} ⊆ ℝ)
143	124	eleq2i 2907	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹}) ↔ 𝑥 ∈ (◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) “ ( E “ {𝐹})))
144		ibar 531	. . . . . . . . . . . . . . . . . 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	119, 144	syl5rbb 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	113, 145	sylan9bb 512	. . . . . . . . . . . . . . . 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	syl5rbb 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 494	. . . . . . . . . . . 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 4055	. . . . . . . . . . 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 3948	. . . . . . . . . . 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 4021	. . . . . . . . . 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 514	. . . . . . . . 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 24091	. . . . . . . . . 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 1114	. . . . . . . . 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 582	. . . . . . . 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 24139	. . . . . . . 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 24141	. . . . . . 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 586	. . . . . 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 1126	. . . . 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 483	. . . 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 2916	. . 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 3185	. 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 24232	. . 3 ⊢ (𝐹:𝐴⟶ℝ → (𝐹 ∈ MblFn ↔ ∀𝑏 ∈ ran (,)(◡𝐹 “ 𝑏) ∈ dom vol))
167	166	3ad2ant1 1129	. 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 398   ∨ wo 843   ∧ w3a 1083   = wceq 1536   ∈ wcel 2113  {cab 2802  ∀wral 3141  ∃wrex 3142  {crab 3145   ∖ cdif 3936   ∪ cun 3937   ∩ cin 3938   ⊆ wss 3939  ∅c0 4294  ifcif 4470  𝒫 cpw 4542  {csn 4570  ∪ ciun 4922   class class class wbr 5069   ↦ cmpt 5149   E cep 5467  ^◡ccnv 5557  dom cdm 5558  ran crn 5559   “ cima 5561   ∘ ccom 5562  Rel wrel 5563   Fn wfn 6353  ⟶wf 6354  ‘cfv 6358  (class class class)co 7159   ↑_m cmap 8409  ℝcr 10539  0cc0 10540  (,)cioo 12741   ↾_t crest 16697  topGenctg 16714  Topctop 21504  TopOnctopon 21521  TopBasesctb 21556   CnP ccnp 21836  vol*covol 24066  volcvol 24067  MblFncmbf 24218

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-inf2 9107  ax-cnex 10596  ax-resscn 10597  ax-1cn 10598  ax-icn 10599  ax-addcl 10600  ax-addrcl 10601  ax-mulcl 10602  ax-mulrcl 10603  ax-mulcom 10604  ax-addass 10605  ax-mulass 10606  ax-distr 10607  ax-i2m1 10608  ax-1ne0 10609  ax-1rid 10610  ax-rnegex 10611  ax-rrecex 10612  ax-cnre 10613  ax-pre-lttri 10614  ax-pre-lttrn 10615  ax-pre-ltadd 10616  ax-pre-mulgt0 10617  ax-pre-sup 10618

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-fal 1549  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-nel 3127  df-ral 3146  df-rex 3147  df-reu 3148  df-rmo 3149  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-int 4880  df-iun 4924  df-disj 5035  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-se 5518  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-isom 6367  df-riota 7117  df-ov 7162  df-oprab 7163  df-mpo 7164  df-of 7412  df-om 7584  df-1st 7692  df-2nd 7693  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-1o 8105  df-2o 8106  df-oadd 8109  df-omul 8110  df-er 8292  df-map 8411  df-pm 8412  df-en 8513  df-dom 8514  df-sdom 8515  df-fin 8516  df-fi 8878  df-sup 8909  df-inf 8910  df-oi 8977  df-dju 9333  df-card 9371  df-acn 9374  df-pnf 10680  df-mnf 10681  df-xr 10682  df-ltxr 10683  df-le 10684  df-sub 10875  df-neg 10876  df-div 11301  df-nn 11642  df-2 11703  df-3 11704  df-4 11705  df-n0 11901  df-z 11985  df-uz 12247  df-q 12352  df-rp 12393  df-xneg 12510  df-xadd 12511  df-xmul 12512  df-ioo 12745  df-ico 12747  df-icc 12748  df-fz 12896  df-fzo 13037  df-fl 13165  df-seq 13373  df-exp 13433  df-hash 13694  df-cj 14461  df-re 14462  df-im 14463  df-sqrt 14597  df-abs 14598  df-clim 14848  df-rlim 14849  df-sum 15046  df-rest 16699  df-topgen 16720  df-psmet 20540  df-xmet 20541  df-met 20542  df-bl 20543  df-mopn 20544  df-top 21505  df-topon 21522  df-bases 21557  df-cnp 21839  df-cmp 21998  df-ovol 24068  df-vol 24069  df-mbf 24223

This theorem is referenced by: (None)