Theorem cnambfre 32628

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 6159	. . . . . . . . 9 ⊢ (𝐹:𝐴⟶ℝ → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
3	2	cnveqd 5220	. . . . . . . 8 ⊢ (𝐹:𝐴⟶ℝ → ◡𝐹 = ◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
4	3	imaeq1d 5384	. . . . . . 7 ⊢ (𝐹:𝐴⟶ℝ → (◡𝐹 “ 𝑏) = (◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) “ 𝑏))
5	4	ad2antrr 758	. . . . . 6 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol) ∧ 𝑏 ∈ ran (,)) → (◡𝐹 “ 𝑏) = (◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) “ 𝑏))
6		exmid 430	. . . . . . . . . . 11 ⊢ (𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∨ ¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥))
7	6	biantrur 526	. . . . . . . . . 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 908	. . . . . . . . . 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 263	. . . . . . . . 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 22374	. . . . . . . . . . . . . . . . . 18 ⊢ ran (,) ∈ TopBases
11		bastg 20581	. . . . . . . . . . . . . . . . . 18 ⊢ (ran (,) ∈ TopBases → ran (,) ⊆ (topGen‘ran (,)))
12	10, 11	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ran (,) ⊆ (topGen‘ran (,))
13	12	sseli 3564	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ∈ ran (,) → 𝑏 ∈ (topGen‘ran (,)))
14	13	ad2antlr 759	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol) ∧ 𝑏 ∈ ran (,)) ∧ 𝑥 ∈ 𝐴) → 𝑏 ∈ (topGen‘ran (,)))
15		cnpimaex 20870	. . . . . . . . . . . . . . . . 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 1261	. . . . . . . . . . . . . . . 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 1257	. . . . . . . . . . . . . . 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 680	. . . . . . . . . . . . . 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 449	. . . . . . . . . . . . 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 801	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol) ∧ (𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))) → (𝐹 “ 𝑦) ⊆ 𝑏)
21		ffn 5958	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹:𝐴⟶ℝ → 𝐹 Fn 𝐴)
22	21	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol) → 𝐹 Fn 𝐴)
23		restsspw 15915	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((topGen‘ran (,)) ↾t 𝐴) ⊆ 𝒫 𝐴
24	23	sseli 3564	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) → 𝑦 ∈ 𝒫 𝐴)
25	24	elpwid 4118	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) → 𝑦 ⊆ 𝐴)
26		simpl 472	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏) → 𝑥 ∈ 𝑦)
27		fnfvima 6400	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑦 ⊆ 𝐴 ∧ 𝑥 ∈ 𝑦) → (𝐹‘𝑥) ∈ (𝐹 “ 𝑦))
28	22, 25, 26, 27	syl3an 1360	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol) ∧ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)) → (𝐹‘𝑥) ∈ (𝐹 “ 𝑦))
29	28	3expb 1258	. . . . . . . . . . . . . . . 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 3569	. . . . . . . . . . . . . . 15 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol) ∧ (𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))) → (𝐹‘𝑥) ∈ 𝑏)
31	30	rexlimdvaa 3014	. . . . . . . . . . . . . 14 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol) → (∃𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)(𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏) → (𝐹‘𝑥) ∈ 𝑏))
32	31	ad3antrrr 762	. . . . . . . . . . . . 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 201	. . . . . . . . . . . 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 671	. . . . . . . . . . 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 3073	. . . . . . . . . . 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 277	. . . . . . . . . 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 735	. . . . . . . . 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 271	. . . . . . . 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 3163	. . . . . . 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 2610	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))
41	40	mptpreima 5545	. . . . . . 7 ⊢ (◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) “ 𝑏) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ∈ 𝑏}
42		unrab 3857	. . . . . . 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 2669	. . . . . 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 2644	. . . . 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 1212	. . . 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 3767	. . . . . . . . 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 3826	. . . . . . . . 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 3858	. . . . . . . . . . . 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 4475	. . . . . . . . . 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 4520	. . . . . . . . . 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 4503	. . . . . . . . . 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 2640	. . . . . . . . 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 2640	. . . . . . . 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 2621	. . . . . . . . . . . . 13 ⊢ (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 2621	. . . . . . . . . . . . 13 ⊢ (∅ = 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 800	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝐹 “ 𝑦) ⊆ 𝑏) ∧ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))) → 𝑥 ∈ 𝑦)
58	25	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝐹 “ 𝑦) ⊆ 𝑏) → 𝑦 ⊆ 𝐴)
59	58	sselda 3568	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝐹 “ 𝑦) ⊆ 𝑏) ∧ 𝑥 ∈ 𝑦) → 𝑥 ∈ 𝐴)
60		pm3.22 464	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 “ 𝑦) ⊆ 𝑏 ∧ 𝑥 ∈ 𝑦) → (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))
61	60	adantll 746	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝐹 “ 𝑦) ⊆ 𝑏) ∧ 𝑥 ∈ 𝑦) → (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))
62	59, 61	jca 553	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝐹 “ 𝑦) ⊆ 𝑏) ∧ 𝑥 ∈ 𝑦) → (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)))
63	57, 62	impbida 873	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝐹 “ 𝑦) ⊆ 𝑏) → ((𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)) ↔ 𝑥 ∈ 𝑦))
64	63	abbidv 2728	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝐹 “ 𝑦) ⊆ 𝑏) → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))} = {𝑥 ∣ 𝑥 ∈ 𝑦})
65		df-rab 2905	. . . . . . . . . . . . . 14 ⊢ {𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))}
66		cvjust 2605	. . . . . . . . . . . . . 14 ⊢ 𝑦 = {𝑥 ∣ 𝑥 ∈ 𝑦}
67	64, 65, 66	3eqtr4g 2669	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ (𝐹 “ 𝑦) ⊆ 𝑏) → {𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} = 𝑦)
68		simpr 476	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏) → (𝐹 “ 𝑦) ⊆ 𝑏)
69	68	con3i 149	. . . . . . . . . . . . . . . 16 ⊢ (¬ (𝐹 “ 𝑦) ⊆ 𝑏 → ¬ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))
70	69	ralrimivw 2950	. . . . . . . . . . . . . . 15 ⊢ (¬ (𝐹 “ 𝑦) ⊆ 𝑏 → ∀𝑥 ∈ 𝐴 ¬ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))
71		rabeq0 3911	. . . . . . . . . . . . . . 15 ⊢ ({𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏))
72	70, 71	sylibr 223	. . . . . . . . . . . . . 14 ⊢ (¬ (𝐹 “ 𝑦) ⊆ 𝑏 → {𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} = ∅)
73	72	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ ¬ (𝐹 “ 𝑦) ⊆ 𝑏) → {𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} = ∅)
74	55, 56, 67, 73	ifbothda 4073	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) → {𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} = if((𝐹 “ 𝑦) ⊆ 𝑏, 𝑦, ∅))
75	74	iuneq2i 4475	. . . . . . . . . . 11 ⊢ ∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} = ∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)if((𝐹 “ 𝑦) ⊆ 𝑏, 𝑦, ∅)
76		retop 22375	. . . . . . . . . . . . . 14 ⊢ (topGen‘ran (,)) ∈ Top
77		resttop 20774	. . . . . . . . . . . . . 14 ⊢ (((topGen‘ran (,)) ∈ Top ∧ 𝐴 ∈ dom vol) → ((topGen‘ran (,)) ↾t 𝐴) ∈ Top)
78	76, 77	mpan 702	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ dom vol → ((topGen‘ran (,)) ↾t 𝐴) ∈ Top)
79		0opn 20534	. . . . . . . . . . . . . . . 16 ⊢ (((topGen‘ran (,)) ↾t 𝐴) ∈ Top → ∅ ∈ ((topGen‘ran (,)) ↾t 𝐴))
80	78, 79	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ dom vol → ∅ ∈ ((topGen‘ran (,)) ↾t 𝐴))
81		ifcl 4080	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ ∅ ∈ ((topGen‘ran (,)) ↾t 𝐴)) → if((𝐹 “ 𝑦) ⊆ 𝑏, 𝑦, ∅) ∈ ((topGen‘ran (,)) ↾t 𝐴))
82	81	ancoms 468	. . . . . . . . . . . . . . 15 ⊢ ((∅ ∈ ((topGen‘ran (,)) ↾t 𝐴) ∧ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)) → if((𝐹 “ 𝑦) ⊆ 𝑏, 𝑦, ∅) ∈ ((topGen‘ran (,)) ↾t 𝐴))
83	80, 82	sylan 487	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ dom vol ∧ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)) → if((𝐹 “ 𝑦) ⊆ 𝑏, 𝑦, ∅) ∈ ((topGen‘ran (,)) ↾t 𝐴))
84	83	ralrimiva 2949	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ dom vol → ∀𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)if((𝐹 “ 𝑦) ⊆ 𝑏, 𝑦, ∅) ∈ ((topGen‘ran (,)) ↾t 𝐴))
85		iunopn 20528	. . . . . . . . . . . . 13 ⊢ ((((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 691	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom vol → ∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)if((𝐹 “ 𝑦) ⊆ 𝑏, 𝑦, ∅) ∈ ((topGen‘ran (,)) ↾t 𝐴))
87		eqid 2610	. . . . . . . . . . . . 13 ⊢ ((topGen‘ran (,)) ↾t 𝐴) = ((topGen‘ran (,)) ↾t 𝐴)
88	87	subopnmbl 23178	. . . . . . . . . . . 12 ⊢ ((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 699	. . . . . . . . . . 11 ⊢ (𝐴 ∈ dom vol → ∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)if((𝐹 “ 𝑦) ⊆ 𝑏, 𝑦, ∅) ∈ dom vol)
90	75, 89	syl5eqel 2692	. . . . . . . . . 10 ⊢ (𝐴 ∈ dom vol → ∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ∈ dom vol)
91	90	adantr 480	. . . . . . . . 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)} ∈ dom vol)
92		difss 3699	. . . . . . . . . . . . 13 ⊢ (∪ 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)}
93		ssrab2 3650	. . . . . . . . . . . . . . 15 ⊢ {𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ⊆ 𝐴
94	93	rgenw 2908	. . . . . . . . . . . . . 14 ⊢ ∀𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ⊆ 𝐴
95		iunss 4497	. . . . . . . . . . . . . 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)} ⊆ 𝐴)
96	94, 95	mpbir 220	. . . . . . . . . . . . 13 ⊢ ∪ 𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴){𝑥 ∈ 𝐴 ∣ (𝑥 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑏)} ⊆ 𝐴
97	92, 96	sstri 3577	. . . . . . . . . . . 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
98		mblss 23106	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ)
99	97, 98	syl5ss 3579	. . . . . . . . . . 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)}) ⊆ ℝ)
100	99	adantr 480	. . . . . . . . . 10 ⊢ ((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)}) ⊆ ℝ)
101		ssdif 3707	. . . . . . . . . . . . . 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)}))
102	96, 101	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)})
103		ovex 6577	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℝ ↑𝑚 𝐴) ∈ V
104	103	rabex 4740	. . . . . . . . . . . . . . . . . . . 20 ⊢ {𝑓 ∈ (ℝ ↑𝑚 𝐴) ∣ ∀𝑏 ∈ (topGen‘ran (,))((𝑓‘𝑥) ∈ 𝑏 → ∃𝑦 ∈ ((topGen‘ran (,)) ↾t 𝐴)(𝑥 ∈ 𝑦 ∧ (𝑓 “ 𝑦) ⊆ 𝑏))} ∈ V
105		eqid 2610	. . . . . . . . . . . . . . . . . . . 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))})
106	104, 105	fnmpti 5935	. . . . . . . . . . . . . . . . . . 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
107		retopon 22377	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
108		resttopon 20775	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ 𝐴 ⊆ ℝ) → ((topGen‘ran (,)) ↾t 𝐴) ∈ (TopOn‘𝐴))
109	107, 98, 108	sylancr 694	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ∈ dom vol → ((topGen‘ran (,)) ↾t 𝐴) ∈ (TopOn‘𝐴))
110		cnpfval 20848	. . . . . . . . . . . . . . . . . . . . 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))}))
111	109, 107, 110	sylancl 693	. . . . . . . . . . . . . . . . . . . 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))}))
112	111	fneq1d 5895	. . . . . . . . . . . . . . . . . . 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))
113	106, 112	mpbiri 247	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ dom vol → (((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) Fn 𝐴)
114		elpreima 6245	. . . . . . . . . . . . . . . . . 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}))))
115	113, 114	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}))))
116		rele 5172	. . . . . . . . . . . . . . . . . . . 20 ⊢ Rel E
117		elrelimasn 5408	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Rel E → (((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∈ ( E “ {𝐹}) ↔ 𝐹 E ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)))
118	116, 117	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∈ ( E “ {𝐹}) ↔ 𝐹 E ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥))
119		fvex 6113	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∈ V
120	119	epelc 4951	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹 E ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ↔ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥))
121	118, 120	bitr2i 264	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ↔ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∈ ( E “ {𝐹}))
122	121	anbi2i 726	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)) ↔ (𝑥 ∈ 𝐴 ∧ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∈ ( E “ {𝐹})))
123	115, 122	syl6rbbr 278	. . . . . . . . . . . . . . . 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}))))
124	123	abbidv 2728	. . . . . . . . . . . . . . 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}))})
125		df-rab 2905	. . . . . . . . . . . . . . 15 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥))}
126		imaco 5557	. . . . . . . . . . . . . . . 16 ⊢ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹}) = (◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) “ ( E “ {𝐹}))
127		abid2 2732	. . . . . . . . . . . . . . . 16 ⊢ {𝑥 ∣ 𝑥 ∈ (◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) “ ( E “ {𝐹}))} = (◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) “ ( E “ {𝐹}))
128	126, 127	eqtr4i 2635	. . . . . . . . . . . . . . 15 ⊢ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹}) = {𝑥 ∣ 𝑥 ∈ (◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) “ ( E “ {𝐹}))}
129	124, 125, 128	3eqtr4g 2669	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ dom vol → {𝑥 ∈ 𝐴 ∣ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)} = ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹}))
130	129	difeq2d 3690	. . . . . . . . . . . . 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})))
131	102, 130	syl5sseq 3616	. . . . . . . . . . . 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})))
132		difss 3699	. . . . . . . . . . . . 13 ⊢ (𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹})) ⊆ 𝐴
133	132, 98	syl5ss 3579	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom vol → (𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹})) ⊆ ℝ)
134	131, 133	jca 553	. . . . . . . . . . 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))
135		ovolssnul 23062	. . . . . . . . . . . 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)
136	135	3expa 1257	. . . . . . . . . . 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)
137	134, 136	sylan 487	. . . . . . . . . 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)
138		nulmbl 23110	. . . . . . . . . 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)
139	100, 137, 138	syl2anc 691	. . . . . . . . 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)
140		difmbl 23118	. . . . . . . . 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)
141	91, 139, 140	syl2anc 691	. . . . . . . 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)
142	54, 141	syl5eqelr 2693	. . . . . . 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)
143		ssrab2 3650	. . . . . . . . . 10 ⊢ {𝑥 ∈ 𝐴 ∣ (¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝑏)} ⊆ 𝐴
144	143, 98	syl5ss 3579	. . . . . . . . 9 ⊢ (𝐴 ∈ dom vol → {𝑥 ∈ 𝐴 ∣ (¬ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝑏)} ⊆ ℝ)
145	144	adantr 480	. . . . . . . 8 ⊢ ((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)} ⊆ ℝ)
146	126	eleq2i 2680	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹}) ↔ 𝑥 ∈ (◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) “ ( E “ {𝐹})))
147		ibar 524	. . . . . . . . . . . . . . . . . 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}))))
148	121, 147	syl5rbb 272	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ∧ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥) ∈ ( E “ {𝐹})) ↔ 𝐹 ∈ ((((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,)))‘𝑥)))
149	115, 148	sylan9bb 732	. . . . . . . . . . . . . . . 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)))
150	146, 149	syl5rbb 272	. . . . . . . . . . . . . . 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})))
151	150	notbid 307	. . . . . . . . . . . . . 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})))
152	151	biimpd 218	. . . . . . . . . . . . 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})))
153	152	adantrd 483	. . . . . . . . . . . 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})))
154	153	ss2rabdv 3646	. . . . . . . . . . 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})})
155		dfdif2 3549	. . . . . . . . . . 11 ⊢ (𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹})) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹})}
156	154, 155	syl6sseqr 3615	. . . . . . . . . 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})))
157	156, 133	jca 553	. . . . . . . . 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))
158		ovolssnul 23062	. . . . . . . . . 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)
159	158	3expa 1257	. . . . . . . . 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)
160	157, 159	sylan 487	. . . . . . . 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)
161		nulmbl 23110	. . . . . . . 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)
162	145, 160, 161	syl2anc 691	. . . . . . 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)
163		unmbl 23112	. . . . . . 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)
164	142, 162, 163	syl2anc 691	. . . . . 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)
165	164	3adant1 1072	. . . . 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)
166	165	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)
167	45, 166	eqeltrd 2688	. . 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)
168	167	ralrimiva 2949	. 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)
169		ismbf 23203	. . 3 ⊢ (𝐹:𝐴⟶ℝ → (𝐹 ∈ MblFn ↔ ∀𝑏 ∈ ran (,)(◡𝐹 “ 𝑏) ∈ dom vol))
170	169	3ad2ant1 1075	. 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))
171	168, 170	mpbird 246	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 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  ∃wrex 2897  {crab 2900   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  ifcif 4036  𝒫 cpw 4108  {csn 4125  ∪ ciun 4455   class class class wbr 4583   ↦ cmpt 4643   E cep 4947  ^◡ccnv 5037  dom cdm 5038  ran crn 5039   “ cima 5041   ∘ ccom 5042  Rel wrel 5043   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑_𝑚 cmap 7744  ℝcr 9814  0cc0 9815  (,)cioo 12046   ↾_t crest 15904  topGenctg 15921  Topctop 20517  TopOnctopon 20518  TopBasesctb 20520   CnP ccnp 20839  vol*covol 23038  volcvol 23039  MblFncmbf 23189

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-omul 7452  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-acn 8651  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-sum 14265  df-rest 15906  df-topgen 15927  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-top 20521  df-bases 20522  df-topon 20523  df-cnp 20842  df-cmp 21000  df-ovol 23040  df-vol 23041  df-mbf 23194

This theorem is referenced by: (None)