Theorem regr1lem2 23764

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 775	. . . . . . . . . 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 776	. . . . . . . . . 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 777	. . . . . . . . . 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 778	. . . . . . . . . 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 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) = ∅))) → 𝑎 ∈ 𝐽)
7		simprr 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) = ∅))) → ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑚 ∧ (𝐹‘𝑤) ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))
8	1, 2, 3, 4, 5, 6, 7	regr1lem 23763	. . . . . . . . 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 1097	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑧) ∈ 𝑚 ∧ (𝐹‘𝑤) ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) ↔ ((𝐹‘𝑤) ∈ 𝑛 ∧ (𝐹‘𝑧) ∈ 𝑚 ∧ (𝑚 ∩ 𝑛) = ∅))
10		incom 4217	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∩ 𝑛) = (𝑛 ∩ 𝑚)
11	10	eqeq1i 2740	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∩ 𝑛) = ∅ ↔ (𝑛 ∩ 𝑚) = ∅)
12	11	3anbi3i 1158	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑤) ∈ 𝑛 ∧ (𝐹‘𝑧) ∈ 𝑚 ∧ (𝑚 ∩ 𝑛) = ∅) ↔ ((𝐹‘𝑤) ∈ 𝑛 ∧ (𝐹‘𝑧) ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))
13	9, 12	bitri 275	. . . . . . . . . . . . 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 3288	. . . . . . . . . . . 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 275	. . . . . . . . . . 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 328	. . . . . . . . . 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 23763	. . . . . . . . 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 212	. . . . . . . 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 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 23748	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ ∀𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝑤 ∈ 𝑦)))
23		elequ2 2121	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑎 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑎))
24		elequ2 2121	. . . . . . . . . . 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 3235	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝑤 ∈ 𝑦) ↔ ∀𝑎 ∈ 𝐽 (𝑧 ∈ 𝑎 ↔ 𝑤 ∈ 𝑎))
27	22, 26	bitrdi 287	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ ∀𝑎 ∈ 𝐽 (𝑧 ∈ 𝑎 ↔ 𝑤 ∈ 𝑎)))
28	27	3expb 1119	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ ∀𝑎 ∈ 𝐽 (𝑧 ∈ 𝑎 ↔ 𝑤 ∈ 𝑎)))
29	28	adantlr 715	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ ∀𝑎 ∈ 𝐽 (𝑧 ∈ 𝑎 ↔ 𝑤 ∈ 𝑎)))
30	21, 29	sylibrd 259	. . . . 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 3200	. . 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 23749	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
34	33	adantr 480	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → 𝐹 Fn 𝑋)
35		neeq1 3001	. . . . . . . 8 ⊢ (𝑎 = (𝐹‘𝑧) → (𝑎 ≠ 𝑏 ↔ (𝐹‘𝑧) ≠ 𝑏))
36		eleq1 2827	. . . . . . . . . 10 ⊢ (𝑎 = (𝐹‘𝑧) → (𝑎 ∈ 𝑚 ↔ (𝐹‘𝑧) ∈ 𝑚))
37	36	3anbi1d 1439	. . . . . . . . 9 ⊢ (𝑎 = (𝐹‘𝑧) → ((𝑎 ∈ 𝑚 ∧ 𝑏 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) ↔ ((𝐹‘𝑧) ∈ 𝑚 ∧ 𝑏 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))
38	37	2rexbidv 3220	. . . . . . . 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 3176	. . . . . 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 7108	. . . . 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 2827	. . . . . . . . . 10 ⊢ (𝑏 = (𝐹‘𝑤) → (𝑏 ∈ 𝑛 ↔ (𝐹‘𝑤) ∈ 𝑛))
44	43	3anbi2d 1440	. . . . . . . . 9 ⊢ (𝑏 = (𝐹‘𝑤) → (((𝐹‘𝑧) ∈ 𝑚 ∧ 𝑏 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) ↔ ((𝐹‘𝑧) ∈ 𝑚 ∧ (𝐹‘𝑤) ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))
45	44	2rexbidv 3220	. . . . . . . 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 7108	. . . . . 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 3176	. . . . 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 279	. . . 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 257	. 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 23751	. . . 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 23375	. . 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 257	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ Haus)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106   ≠ wne 2938  ∀wral 3059  ∃wrex 3068  {crab 3433   ∩ cin 3962  ∅c0 4339   ↦ cmpt 5231  ran crn 5690   Fn wfn 6558  ‘cfv 6563  TopOnctopon 22932  Hauscha 23332  Regcreg 23333  KQckq 23717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-qtop 17554  df-top 22916  df-topon 22933  df-cld 23043  df-cls 23045  df-haus 23339  df-reg 23340  df-kq 23718

This theorem is referenced by:  regr1  23774