Theorem regr1lem2 23897

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 786	. . . . . . . . . 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 787	. . . . . . . . . 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 788	. . . . . . . . . 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 789	. . . . . . . . . 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 782	. . . . . . . . . 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 784	. . . . . . . . . 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 23896	. . . . . . . . 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 1115	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑧) ∈ 𝑚 ∧ (𝐹‘𝑤) ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) ↔ ((𝐹‘𝑤) ∈ 𝑛 ∧ (𝐹‘𝑧) ∈ 𝑚 ∧ (𝑚 ∩ 𝑛) = ∅))
10		incom 4162	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∩ 𝑛) = (𝑛 ∩ 𝑚)
11	10	eqeq1i 2768	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∩ 𝑛) = ∅ ↔ (𝑛 ∩ 𝑚) = ∅)
12	11	3anbi3i 1177	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑤) ∈ 𝑛 ∧ (𝐹‘𝑧) ∈ 𝑚 ∧ (𝑚 ∩ 𝑛) = ∅) ↔ ((𝐹‘𝑤) ∈ 𝑛 ∧ (𝐹‘𝑧) ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))
13	9, 12	bitri 278	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑧) ∈ 𝑚 ∧ (𝐹‘𝑤) ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) ↔ ((𝐹‘𝑤) ∈ 𝑛 ∧ (𝐹‘𝑧) ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))
14	13	2rexbii 3141	. . . . . . . . . . . 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 3294	. . . . . . . . . . . 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 278	. . . . . . . . . . 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 331	. . . . . . . . . 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 23896	. . . . . . . . 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 215	. . . . . . . 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 461	. . . . . . 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 3165	. . . . . 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 23881	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ ∀𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝑤 ∈ 𝑦)))
23		elequ2 2158	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑎 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑎))
24		elequ2 2158	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑎 → (𝑤 ∈ 𝑦 ↔ 𝑤 ∈ 𝑎))
25	23, 24	bibi12d 348	. . . . . . . . . 10 ⊢ (𝑦 = 𝑎 → ((𝑧 ∈ 𝑦 ↔ 𝑤 ∈ 𝑦) ↔ (𝑧 ∈ 𝑎 ↔ 𝑤 ∈ 𝑎)))
26	25	cbvralvw 3243	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝑤 ∈ 𝑦) ↔ ∀𝑎 ∈ 𝐽 (𝑧 ∈ 𝑎 ↔ 𝑤 ∈ 𝑎))
27	22, 26	bitrdi 290	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ ∀𝑎 ∈ 𝐽 (𝑧 ∈ 𝑎 ↔ 𝑤 ∈ 𝑎)))
28	27	3expb 1138	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ ∀𝑎 ∈ 𝐽 (𝑧 ∈ 𝑎 ↔ 𝑤 ∈ 𝑎)))
29	28	adantlr 727	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ ∀𝑎 ∈ 𝐽 (𝑧 ∈ 𝑎 ↔ 𝑤 ∈ 𝑎)))
30	21, 29	sylibrd 262	. . . . 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 2975	. . . 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 3208	. . 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 23882	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
34	33	adantr 485	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → 𝐹 Fn 𝑋)
35		neeq1 3020	. . . . . . . 8 ⊢ (𝑎 = (𝐹‘𝑧) → (𝑎 ≠ 𝑏 ↔ (𝐹‘𝑧) ≠ 𝑏))
36		eleq1 2851	. . . . . . . . . 10 ⊢ (𝑎 = (𝐹‘𝑧) → (𝑎 ∈ 𝑚 ↔ (𝐹‘𝑧) ∈ 𝑚))
37	36	3anbi1d 1468	. . . . . . . . 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 347	. . . . . . 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 3188	. . . . . 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 7083	. . . . 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 3021	. . . . . . . 8 ⊢ (𝑏 = (𝐹‘𝑤) → ((𝐹‘𝑧) ≠ 𝑏 ↔ (𝐹‘𝑧) ≠ (𝐹‘𝑤)))
43		eleq1 2851	. . . . . . . . . 10 ⊢ (𝑏 = (𝐹‘𝑤) → (𝑏 ∈ 𝑛 ↔ (𝐹‘𝑤) ∈ 𝑛))
44	43	3anbi2d 1469	. . . . . . . . 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 347	. . . . . . 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 7083	. . . . . 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 3188	. . . . 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 282	. . . 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 18	. . 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 260	. 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 23884	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
53	52	adantr 485	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
54		ishaus2 23508	. . 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 18	. 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 260	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ Haus)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  ∃wrex 3089  {crab 3416   ∩ cin 3904  ∅c0 4286   ↦ cmpt 5192  ran crn 5662   Fn wfn 6531  ‘cfv 6536  TopOnctopon 23067  Hauscha 23465  Regcreg 23466  KQckq 23850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-iin 4959  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-qtop 17556  df-top 23051  df-topon 23068  df-cld 23176  df-cls 23178  df-haus 23472  df-reg 23473  df-kq 23851

This theorem is referenced by:  regr1  23907