Theorem regr1lem2 22872

Description: A Kolmogorov quotient of a regular space is Hausdorff. (Contributed by Mario Carneiro, 25-Aug-2015.)

Hypothesis

Ref	Expression
kqval.2	⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})

Assertion

Ref	Expression
regr1lem2	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ Haus)

Distinct variable groups:   𝑥,𝑦,𝐽   𝑥,𝑋,𝑦

Allowed substitution hints:   𝐹(𝑥,𝑦)

Proof of Theorem regr1lem2

Dummy variables 𝑚 𝑛 𝑤 𝑧 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		kqval.2	. . . . . . . . . 10 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
2		simplll 771	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑎 ∈ 𝐽 ∧ ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑚 ∧ (𝐹‘𝑤) ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))) → 𝐽 ∈ (TopOn‘𝑋))
3		simpllr 772	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑎 ∈ 𝐽 ∧ ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑚 ∧ (𝐹‘𝑤) ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))) → 𝐽 ∈ Reg)
4		simplrl 773	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑎 ∈ 𝐽 ∧ ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑚 ∧ (𝐹‘𝑤) ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))) → 𝑧 ∈ 𝑋)
5		simplrr 774	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑎 ∈ 𝐽 ∧ ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑚 ∧ (𝐹‘𝑤) ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))) → 𝑤 ∈ 𝑋)
6		simprl 767	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑎 ∈ 𝐽 ∧ ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑚 ∧ (𝐹‘𝑤) ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))) → 𝑎 ∈ 𝐽)
7		simprr 769	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑎 ∈ 𝐽 ∧ ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑚 ∧ (𝐹‘𝑤) ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))) → ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑚 ∧ (𝐹‘𝑤) ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))
8	1, 2, 3, 4, 5, 6, 7	regr1lem 22871	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑎 ∈ 𝐽 ∧ ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑚 ∧ (𝐹‘𝑤) ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))) → (𝑧 ∈ 𝑎 → 𝑤 ∈ 𝑎))
9		3ancoma 1096	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑧) ∈ 𝑚 ∧ (𝐹‘𝑤) ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) ↔ ((𝐹‘𝑤) ∈ 𝑛 ∧ (𝐹‘𝑧) ∈ 𝑚 ∧ (𝑚 ∩ 𝑛) = ∅))
10		incom 4139	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∩ 𝑛) = (𝑛 ∩ 𝑚)
11	10	eqeq1i 2744	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∩ 𝑛) = ∅ ↔ (𝑛 ∩ 𝑚) = ∅)
12	11	3anbi3i 1157	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑤) ∈ 𝑛 ∧ (𝐹‘𝑧) ∈ 𝑚 ∧ (𝑚 ∩ 𝑛) = ∅) ↔ ((𝐹‘𝑤) ∈ 𝑛 ∧ (𝐹‘𝑧) ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))
13	9, 12	bitri 274	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑧) ∈ 𝑚 ∧ (𝐹‘𝑤) ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) ↔ ((𝐹‘𝑤) ∈ 𝑛 ∧ (𝐹‘𝑧) ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))
14	13	2rexbii 3180	. . . . . . . . . . . 12 ⊢ (∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑚 ∧ (𝐹‘𝑤) ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) ↔ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝑤) ∈ 𝑛 ∧ (𝐹‘𝑧) ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))
15		rexcom 3283	. . . . . . . . . . . 12 ⊢ (∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝑤) ∈ 𝑛 ∧ (𝐹‘𝑧) ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅) ↔ ∃𝑛 ∈ (KQ‘𝐽)∃𝑚 ∈ (KQ‘𝐽)((𝐹‘𝑤) ∈ 𝑛 ∧ (𝐹‘𝑧) ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))
16	14, 15	bitri 274	. . . . . . . . . . 11 ⊢ (∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑚 ∧ (𝐹‘𝑤) ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) ↔ ∃𝑛 ∈ (KQ‘𝐽)∃𝑚 ∈ (KQ‘𝐽)((𝐹‘𝑤) ∈ 𝑛 ∧ (𝐹‘𝑧) ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))
17	7, 16	sylnib 327	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑎 ∈ 𝐽 ∧ ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑚 ∧ (𝐹‘𝑤) ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))) → ¬ ∃𝑛 ∈ (KQ‘𝐽)∃𝑚 ∈ (KQ‘𝐽)((𝐹‘𝑤) ∈ 𝑛 ∧ (𝐹‘𝑧) ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))
18	1, 2, 3, 5, 4, 6, 17	regr1lem 22871	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑎 ∈ 𝐽 ∧ ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑚 ∧ (𝐹‘𝑤) ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))) → (𝑤 ∈ 𝑎 → 𝑧 ∈ 𝑎))
19	8, 18	impbid 211	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑎 ∈ 𝐽 ∧ ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑚 ∧ (𝐹‘𝑤) ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))) → (𝑧 ∈ 𝑎 ↔ 𝑤 ∈ 𝑎))
20	19	expr 456	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑎 ∈ 𝐽) → (¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑚 ∧ (𝐹‘𝑤) ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) → (𝑧 ∈ 𝑎 ↔ 𝑤 ∈ 𝑎)))
21	20	ralrimdva 3114	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑚 ∧ (𝐹‘𝑤) ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) → ∀𝑎 ∈ 𝐽 (𝑧 ∈ 𝑎 ↔ 𝑤 ∈ 𝑎)))
22	1	kqfeq 22856	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ ∀𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝑤 ∈ 𝑦)))
23		elequ2 2124	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑎 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑎))
24		elequ2 2124	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑎 → (𝑤 ∈ 𝑦 ↔ 𝑤 ∈ 𝑎))
25	23, 24	bibi12d 345	. . . . . . . . . 10 ⊢ (𝑦 = 𝑎 → ((𝑧 ∈ 𝑦 ↔ 𝑤 ∈ 𝑦) ↔ (𝑧 ∈ 𝑎 ↔ 𝑤 ∈ 𝑎)))
26	25	cbvralvw 3380	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝑤 ∈ 𝑦) ↔ ∀𝑎 ∈ 𝐽 (𝑧 ∈ 𝑎 ↔ 𝑤 ∈ 𝑎))
27	22, 26	bitrdi 286	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ ∀𝑎 ∈ 𝐽 (𝑧 ∈ 𝑎 ↔ 𝑤 ∈ 𝑎)))
28	27	3expb 1118	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ ∀𝑎 ∈ 𝐽 (𝑧 ∈ 𝑎 ↔ 𝑤 ∈ 𝑎)))
29	28	adantlr 711	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ ∀𝑎 ∈ 𝐽 (𝑧 ∈ 𝑎 ↔ 𝑤 ∈ 𝑎)))
30	21, 29	sylibrd 258	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑚 ∧ (𝐹‘𝑤) ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) → (𝐹‘𝑧) = (𝐹‘𝑤)))
31	30	necon1ad 2961	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝐹‘𝑧) ≠ (𝐹‘𝑤) → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑚 ∧ (𝐹‘𝑤) ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))
32	31	ralrimivva 3116	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝐹‘𝑧) ≠ (𝐹‘𝑤) → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑚 ∧ (𝐹‘𝑤) ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))
33	1	kqffn 22857	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
34	33	adantr 480	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → 𝐹 Fn 𝑋)
35		neeq1 3007	. . . . . . . 8 ⊢ (𝑎 = (𝐹‘𝑧) → (𝑎 ≠ 𝑏 ↔ (𝐹‘𝑧) ≠ 𝑏))
36		eleq1 2827	. . . . . . . . . 10 ⊢ (𝑎 = (𝐹‘𝑧) → (𝑎 ∈ 𝑚 ↔ (𝐹‘𝑧) ∈ 𝑚))
37	36	3anbi1d 1438	. . . . . . . . 9 ⊢ (𝑎 = (𝐹‘𝑧) → ((𝑎 ∈ 𝑚 ∧ 𝑏 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) ↔ ((𝐹‘𝑧) ∈ 𝑚 ∧ 𝑏 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))
38	37	2rexbidv 3230	. . . . . . . 8 ⊢ (𝑎 = (𝐹‘𝑧) → (∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)(𝑎 ∈ 𝑚 ∧ 𝑏 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) ↔ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑚 ∧ 𝑏 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))
39	35, 38	imbi12d 344	. . . . . . 7 ⊢ (𝑎 = (𝐹‘𝑧) → ((𝑎 ≠ 𝑏 → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)(𝑎 ∈ 𝑚 ∧ 𝑏 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)) ↔ ((𝐹‘𝑧) ≠ 𝑏 → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑚 ∧ 𝑏 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))))
40	39	ralbidv 3122	. . . . . 6 ⊢ (𝑎 = (𝐹‘𝑧) → (∀𝑏 ∈ ran 𝐹(𝑎 ≠ 𝑏 → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)(𝑎 ∈ 𝑚 ∧ 𝑏 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)) ↔ ∀𝑏 ∈ ran 𝐹((𝐹‘𝑧) ≠ 𝑏 → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑚 ∧ 𝑏 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))))
41	40	ralrn 6958	. . . . 5 ⊢ (𝐹 Fn 𝑋 → (∀𝑎 ∈ ran 𝐹∀𝑏 ∈ ran 𝐹(𝑎 ≠ 𝑏 → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)(𝑎 ∈ 𝑚 ∧ 𝑏 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)) ↔ ∀𝑧 ∈ 𝑋 ∀𝑏 ∈ ran 𝐹((𝐹‘𝑧) ≠ 𝑏 → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑚 ∧ 𝑏 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))))
42		neeq2 3008	. . . . . . . 8 ⊢ (𝑏 = (𝐹‘𝑤) → ((𝐹‘𝑧) ≠ 𝑏 ↔ (𝐹‘𝑧) ≠ (𝐹‘𝑤)))
43		eleq1 2827	. . . . . . . . . 10 ⊢ (𝑏 = (𝐹‘𝑤) → (𝑏 ∈ 𝑛 ↔ (𝐹‘𝑤) ∈ 𝑛))
44	43	3anbi2d 1439	. . . . . . . . 9 ⊢ (𝑏 = (𝐹‘𝑤) → (((𝐹‘𝑧) ∈ 𝑚 ∧ 𝑏 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) ↔ ((𝐹‘𝑧) ∈ 𝑚 ∧ (𝐹‘𝑤) ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))
45	44	2rexbidv 3230	. . . . . . . 8 ⊢ (𝑏 = (𝐹‘𝑤) → (∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑚 ∧ 𝑏 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) ↔ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑚 ∧ (𝐹‘𝑤) ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))
46	42, 45	imbi12d 344	. . . . . . 7 ⊢ (𝑏 = (𝐹‘𝑤) → (((𝐹‘𝑧) ≠ 𝑏 → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑚 ∧ 𝑏 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)) ↔ ((𝐹‘𝑧) ≠ (𝐹‘𝑤) → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑚 ∧ (𝐹‘𝑤) ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))))
47	46	ralrn 6958	. . . . . 6 ⊢ (𝐹 Fn 𝑋 → (∀𝑏 ∈ ran 𝐹((𝐹‘𝑧) ≠ 𝑏 → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑚 ∧ 𝑏 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)) ↔ ∀𝑤 ∈ 𝑋 ((𝐹‘𝑧) ≠ (𝐹‘𝑤) → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑚 ∧ (𝐹‘𝑤) ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))))
48	47	ralbidv 3122	. . . . 5 ⊢ (𝐹 Fn 𝑋 → (∀𝑧 ∈ 𝑋 ∀𝑏 ∈ ran 𝐹((𝐹‘𝑧) ≠ 𝑏 → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑚 ∧ 𝑏 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)) ↔ ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝐹‘𝑧) ≠ (𝐹‘𝑤) → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑚 ∧ (𝐹‘𝑤) ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))))
49	41, 48	bitrd 278	. . . 4 ⊢ (𝐹 Fn 𝑋 → (∀𝑎 ∈ ran 𝐹∀𝑏 ∈ ran 𝐹(𝑎 ≠ 𝑏 → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)(𝑎 ∈ 𝑚 ∧ 𝑏 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)) ↔ ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝐹‘𝑧) ≠ (𝐹‘𝑤) → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑚 ∧ (𝐹‘𝑤) ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))))
50	34, 49	syl 17	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → (∀𝑎 ∈ ran 𝐹∀𝑏 ∈ ran 𝐹(𝑎 ≠ 𝑏 → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)(𝑎 ∈ 𝑚 ∧ 𝑏 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)) ↔ ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝐹‘𝑧) ≠ (𝐹‘𝑤) → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑚 ∧ (𝐹‘𝑤) ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))))
51	32, 50	mpbird 256	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → ∀𝑎 ∈ ran 𝐹∀𝑏 ∈ ran 𝐹(𝑎 ≠ 𝑏 → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)(𝑎 ∈ 𝑚 ∧ 𝑏 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))
52	1	kqtopon 22859	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
53	52	adantr 480	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
54		ishaus2 22483	. . 3 ⊢ ((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) → ((KQ‘𝐽) ∈ Haus ↔ ∀𝑎 ∈ ran 𝐹∀𝑏 ∈ ran 𝐹(𝑎 ≠ 𝑏 → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)(𝑎 ∈ 𝑚 ∧ 𝑏 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))))
55	53, 54	syl 17	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → ((KQ‘𝐽) ∈ Haus ↔ ∀𝑎 ∈ ran 𝐹∀𝑏 ∈ ran 𝐹(𝑎 ≠ 𝑏 → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)(𝑎 ∈ 𝑚 ∧ 𝑏 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))))
56	51, 55	mpbird 256	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ Haus)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1541   ∈ wcel 2109   ≠ wne 2944  ∀wral 3065  ∃wrex 3066  {crab 3069   ∩ cin 3890  ∅c0 4261   ↦ cmpt 5161  ran crn 5589   Fn wfn 6425  ‘cfv 6430  TopOnctopon 22040  Hauscha 22440  Regcreg 22441  KQckq 22825

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3072  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-int 4885  df-iun 4931  df-iin 4932  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-ov 7271  df-oprab 7272  df-mpo 7273  df-qtop 17199  df-top 22024  df-topon 22041  df-cld 22151  df-cls 22153  df-haus 22447  df-reg 22448  df-kq 22826

This theorem is referenced by:  regr1  22882