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Theorem regr1lem2 23464

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 23463	. . . . . . . . 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 4200	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∩ 𝑛) = (𝑛 ∩ 𝑚)
11	10	eqeq1i 2735	. . . . . . . . . . . . . . 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 3127	. . . . . . . . . . . 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 3285	. . . . . . . . . . . 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 23463	. . . . . . . . 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 455	. . . . . . 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 3152	. . . . . 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 23448	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ ∀𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝑤 ∈ 𝑦)))
23		elequ2 2119	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑎 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑎))
24		elequ2 2119	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑎 → (𝑤 ∈ 𝑦 ↔ 𝑤 ∈ 𝑎))
25	23, 24	bibi12d 344	. . . . . . . . . 10 ⊢ (𝑦 = 𝑎 → ((𝑧 ∈ 𝑦 ↔ 𝑤 ∈ 𝑦) ↔ (𝑧 ∈ 𝑎 ↔ 𝑤 ∈ 𝑎)))
26	25	cbvralvw 3232	. . . . . . . . 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 2955	. . . 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 3198	. . 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 23449	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
34	33	adantr 479	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → 𝐹 Fn 𝑋)
35		neeq1 3001	. . . . . . . 8 ⊢ (𝑎 = (𝐹‘𝑧) → (𝑎 ≠ 𝑏 ↔ (𝐹‘𝑧) ≠ 𝑏))
36		eleq1 2819	. . . . . . . . . 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 3217	. . . . . . . 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 343	. . . . . . 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 3175	. . . . . 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 7088	. . . . 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 3002	. . . . . . . 8 ⊢ (𝑏 = (𝐹‘𝑤) → ((𝐹‘𝑧) ≠ 𝑏 ↔ (𝐹‘𝑧) ≠ (𝐹‘𝑤)))
43		eleq1 2819	. . . . . . . . . 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 3217	. . . . . . . 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 343	. . . . . . 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 7088	. . . . . 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 3175	. . . . 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 23451	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
53	52	adantr 479	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
54		ishaus2 23075	. . 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)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 ∀wral 3059 ∃wrex 3068 {crab 3430 ∩ cin 3946 ∅c0 4321 ↦ cmpt 5230 ran crn 5676 Fn wfn 6537 ‘cfv 6542 TopOnctopon 22632 Hauscha 23032 Regcreg 23033 KQckq 23417

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7414 df-oprab 7415 df-mpo 7416 df-qtop 17457 df-top 22616 df-topon 22633 df-cld 22743 df-cls 22745 df-haus 23039 df-reg 23040 df-kq 23418

This theorem is referenced by: regr1 23474

W3C validator