Theorem regr1lem2 23664

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 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) = ∅))) → 𝐽 ∈ (TopOn‘𝑋))
3		simpllr 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) = ∅))) → 𝐽 ∈ Reg)
4		simplrl 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) = ∅))) → 𝑧 ∈ 𝑋)
5		simplrr 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) = ∅))) → 𝑤 ∈ 𝑋)
6		simprl 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) = ∅))) → 𝑎 ∈ 𝐽)
7		simprr 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) = ∅))) → ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑚 ∧ (𝐹‘𝑤) ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))
8	1, 2, 3, 4, 5, 6, 7	regr1lem 23663	. . . . . . . . 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 1095	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑧) ∈ 𝑚 ∧ (𝐹‘𝑤) ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) ↔ ((𝐹‘𝑤) ∈ 𝑛 ∧ (𝐹‘𝑧) ∈ 𝑚 ∧ (𝑚 ∩ 𝑛) = ∅))
10		incom 4203	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∩ 𝑛) = (𝑛 ∩ 𝑚)
11	10	eqeq1i 2733	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∩ 𝑛) = ∅ ↔ (𝑛 ∩ 𝑚) = ∅)
12	11	3anbi3i 1156	. . . . . . . . . . . . . 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 3126	. . . . . . . . . . . 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 23663	. . . . . . . . 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 3151	. . . . . 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 23648	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ ∀𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝑤 ∈ 𝑦)))
23		elequ2 2113	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑎 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑎))
24		elequ2 2113	. . . . . . . . . . 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 1117	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ ∀𝑎 ∈ 𝐽 (𝑧 ∈ 𝑎 ↔ 𝑤 ∈ 𝑎)))
29	28	adantlr 713	. . . . . 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 2954	. . . 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 23649	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
34	33	adantr 479	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → 𝐹 Fn 𝑋)
35		neeq1 3000	. . . . . . . 8 ⊢ (𝑎 = (𝐹‘𝑧) → (𝑎 ≠ 𝑏 ↔ (𝐹‘𝑧) ≠ 𝑏))
36		eleq1 2817	. . . . . . . . . 10 ⊢ (𝑎 = (𝐹‘𝑧) → (𝑎 ∈ 𝑚 ↔ (𝐹‘𝑧) ∈ 𝑚))
37	36	3anbi1d 1436	. . . . . . . . 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 7103	. . . . 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 3001	. . . . . . . 8 ⊢ (𝑏 = (𝐹‘𝑤) → ((𝐹‘𝑧) ≠ 𝑏 ↔ (𝐹‘𝑧) ≠ (𝐹‘𝑤)))
43		eleq1 2817	. . . . . . . . . 10 ⊢ (𝑏 = (𝐹‘𝑤) → (𝑏 ∈ 𝑛 ↔ (𝐹‘𝑤) ∈ 𝑛))
44	43	3anbi2d 1437	. . . . . . . . 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 7103	. . . . . 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 23651	. . . 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 23275	. . 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ≠ wne 2937  ∀wral 3058  ∃wrex 3067  {crab 3430   ∩ cin 3948  ∅c0 4326   ↦ cmpt 5235  ran crn 5683   Fn wfn 6548  ‘cfv 6553  TopOnctopon 22832  Hauscha 23232  Regcreg 23233  KQckq 23617

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-iin 5003  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-qtop 17496  df-top 22816  df-topon 22833  df-cld 22943  df-cls 22945  df-haus 23239  df-reg 23240  df-kq 23618

This theorem is referenced by:  regr1  23674