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Theorem regr1lem2 22336
 Description: A Kolmogorov quotient of a regular space is Hausdorff. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
kqval.2 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
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.
StepHypRef Expression
1 kqval.2 . . . . . . . . . 10 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
2 simplll 774 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧𝑋𝑤𝑋)) ∧ (𝑎𝐽 ∧ ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅))) → 𝐽 ∈ (TopOn‘𝑋))
3 simpllr 775 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧𝑋𝑤𝑋)) ∧ (𝑎𝐽 ∧ ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅))) → 𝐽 ∈ Reg)
4 simplrl 776 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧𝑋𝑤𝑋)) ∧ (𝑎𝐽 ∧ ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅))) → 𝑧𝑋)
5 simplrr 777 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧𝑋𝑤𝑋)) ∧ (𝑎𝐽 ∧ ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅))) → 𝑤𝑋)
6 simprl 770 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧𝑋𝑤𝑋)) ∧ (𝑎𝐽 ∧ ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅))) → 𝑎𝐽)
7 simprr 772 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧𝑋𝑤𝑋)) ∧ (𝑎𝐽 ∧ ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅))) → ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅))
81, 2, 3, 4, 5, 6, 7regr1lem 22335 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧𝑋𝑤𝑋)) ∧ (𝑎𝐽 ∧ ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅))) → (𝑧𝑎𝑤𝑎))
9 3ancoma 1095 . . . . . . . . . . . . . 14 (((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅) ↔ ((𝐹𝑤) ∈ 𝑛 ∧ (𝐹𝑧) ∈ 𝑚 ∧ (𝑚𝑛) = ∅))
10 incom 4161 . . . . . . . . . . . . . . . 16 (𝑚𝑛) = (𝑛𝑚)
1110eqeq1i 2829 . . . . . . . . . . . . . . 15 ((𝑚𝑛) = ∅ ↔ (𝑛𝑚) = ∅)
12113anbi3i 1156 . . . . . . . . . . . . . 14 (((𝐹𝑤) ∈ 𝑛 ∧ (𝐹𝑧) ∈ 𝑚 ∧ (𝑚𝑛) = ∅) ↔ ((𝐹𝑤) ∈ 𝑛 ∧ (𝐹𝑧) ∈ 𝑚 ∧ (𝑛𝑚) = ∅))
139, 12bitri 278 . . . . . . . . . . . . 13 (((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅) ↔ ((𝐹𝑤) ∈ 𝑛 ∧ (𝐹𝑧) ∈ 𝑚 ∧ (𝑛𝑚) = ∅))
14132rexbii 3242 . . . . . . . . . . . 12 (∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅) ↔ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑤) ∈ 𝑛 ∧ (𝐹𝑧) ∈ 𝑚 ∧ (𝑛𝑚) = ∅))
15 rexcom 3346 . . . . . . . . . . . 12 (∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑤) ∈ 𝑛 ∧ (𝐹𝑧) ∈ 𝑚 ∧ (𝑛𝑚) = ∅) ↔ ∃𝑛 ∈ (KQ‘𝐽)∃𝑚 ∈ (KQ‘𝐽)((𝐹𝑤) ∈ 𝑛 ∧ (𝐹𝑧) ∈ 𝑚 ∧ (𝑛𝑚) = ∅))
1614, 15bitri 278 . . . . . . . . . . 11 (∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅) ↔ ∃𝑛 ∈ (KQ‘𝐽)∃𝑚 ∈ (KQ‘𝐽)((𝐹𝑤) ∈ 𝑛 ∧ (𝐹𝑧) ∈ 𝑚 ∧ (𝑛𝑚) = ∅))
177, 16sylnib 331 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧𝑋𝑤𝑋)) ∧ (𝑎𝐽 ∧ ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅))) → ¬ ∃𝑛 ∈ (KQ‘𝐽)∃𝑚 ∈ (KQ‘𝐽)((𝐹𝑤) ∈ 𝑛 ∧ (𝐹𝑧) ∈ 𝑚 ∧ (𝑛𝑚) = ∅))
181, 2, 3, 5, 4, 6, 17regr1lem 22335 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧𝑋𝑤𝑋)) ∧ (𝑎𝐽 ∧ ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅))) → (𝑤𝑎𝑧𝑎))
198, 18impbid 215 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧𝑋𝑤𝑋)) ∧ (𝑎𝐽 ∧ ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅))) → (𝑧𝑎𝑤𝑎))
2019expr 460 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑎𝐽) → (¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅) → (𝑧𝑎𝑤𝑎)))
2120ralrimdva 3183 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧𝑋𝑤𝑋)) → (¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅) → ∀𝑎𝐽 (𝑧𝑎𝑤𝑎)))
221kqfeq 22320 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋𝑤𝑋) → ((𝐹𝑧) = (𝐹𝑤) ↔ ∀𝑦𝐽 (𝑧𝑦𝑤𝑦)))
23 elequ2 2130 . . . . . . . . . . 11 (𝑦 = 𝑎 → (𝑧𝑦𝑧𝑎))
24 elequ2 2130 . . . . . . . . . . 11 (𝑦 = 𝑎 → (𝑤𝑦𝑤𝑎))
2523, 24bibi12d 349 . . . . . . . . . 10 (𝑦 = 𝑎 → ((𝑧𝑦𝑤𝑦) ↔ (𝑧𝑎𝑤𝑎)))
2625cbvralvw 3434 . . . . . . . . 9 (∀𝑦𝐽 (𝑧𝑦𝑤𝑦) ↔ ∀𝑎𝐽 (𝑧𝑎𝑤𝑎))
2722, 26syl6bb 290 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋𝑤𝑋) → ((𝐹𝑧) = (𝐹𝑤) ↔ ∀𝑎𝐽 (𝑧𝑎𝑤𝑎)))
28273expb 1117 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑧𝑋𝑤𝑋)) → ((𝐹𝑧) = (𝐹𝑤) ↔ ∀𝑎𝐽 (𝑧𝑎𝑤𝑎)))
2928adantlr 714 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧𝑋𝑤𝑋)) → ((𝐹𝑧) = (𝐹𝑤) ↔ ∀𝑎𝐽 (𝑧𝑎𝑤𝑎)))
3021, 29sylibrd 262 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧𝑋𝑤𝑋)) → (¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅) → (𝐹𝑧) = (𝐹𝑤)))
3130necon1ad 3030 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧𝑋𝑤𝑋)) → ((𝐹𝑧) ≠ (𝐹𝑤) → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅)))
3231ralrimivva 3185 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → ∀𝑧𝑋𝑤𝑋 ((𝐹𝑧) ≠ (𝐹𝑤) → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅)))
331kqffn 22321 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
3433adantr 484 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → 𝐹 Fn 𝑋)
35 neeq1 3075 . . . . . . . 8 (𝑎 = (𝐹𝑧) → (𝑎𝑏 ↔ (𝐹𝑧) ≠ 𝑏))
36 eleq1 2903 . . . . . . . . . 10 (𝑎 = (𝐹𝑧) → (𝑎𝑚 ↔ (𝐹𝑧) ∈ 𝑚))
37363anbi1d 1437 . . . . . . . . 9 (𝑎 = (𝐹𝑧) → ((𝑎𝑚𝑏𝑛 ∧ (𝑚𝑛) = ∅) ↔ ((𝐹𝑧) ∈ 𝑚𝑏𝑛 ∧ (𝑚𝑛) = ∅)))
38372rexbidv 3292 . . . . . . . 8 (𝑎 = (𝐹𝑧) → (∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)(𝑎𝑚𝑏𝑛 ∧ (𝑚𝑛) = ∅) ↔ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚𝑏𝑛 ∧ (𝑚𝑛) = ∅)))
3935, 38imbi12d 348 . . . . . . 7 (𝑎 = (𝐹𝑧) → ((𝑎𝑏 → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)(𝑎𝑚𝑏𝑛 ∧ (𝑚𝑛) = ∅)) ↔ ((𝐹𝑧) ≠ 𝑏 → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚𝑏𝑛 ∧ (𝑚𝑛) = ∅))))
4039ralbidv 3191 . . . . . 6 (𝑎 = (𝐹𝑧) → (∀𝑏 ∈ ran 𝐹(𝑎𝑏 → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)(𝑎𝑚𝑏𝑛 ∧ (𝑚𝑛) = ∅)) ↔ ∀𝑏 ∈ ran 𝐹((𝐹𝑧) ≠ 𝑏 → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚𝑏𝑛 ∧ (𝑚𝑛) = ∅))))
4140ralrn 6837 . . . . 5 (𝐹 Fn 𝑋 → (∀𝑎 ∈ ran 𝐹𝑏 ∈ ran 𝐹(𝑎𝑏 → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)(𝑎𝑚𝑏𝑛 ∧ (𝑚𝑛) = ∅)) ↔ ∀𝑧𝑋𝑏 ∈ ran 𝐹((𝐹𝑧) ≠ 𝑏 → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚𝑏𝑛 ∧ (𝑚𝑛) = ∅))))
42 neeq2 3076 . . . . . . . 8 (𝑏 = (𝐹𝑤) → ((𝐹𝑧) ≠ 𝑏 ↔ (𝐹𝑧) ≠ (𝐹𝑤)))
43 eleq1 2903 . . . . . . . . . 10 (𝑏 = (𝐹𝑤) → (𝑏𝑛 ↔ (𝐹𝑤) ∈ 𝑛))
44433anbi2d 1438 . . . . . . . . 9 (𝑏 = (𝐹𝑤) → (((𝐹𝑧) ∈ 𝑚𝑏𝑛 ∧ (𝑚𝑛) = ∅) ↔ ((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅)))
45442rexbidv 3292 . . . . . . . 8 (𝑏 = (𝐹𝑤) → (∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚𝑏𝑛 ∧ (𝑚𝑛) = ∅) ↔ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅)))
4642, 45imbi12d 348 . . . . . . 7 (𝑏 = (𝐹𝑤) → (((𝐹𝑧) ≠ 𝑏 → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚𝑏𝑛 ∧ (𝑚𝑛) = ∅)) ↔ ((𝐹𝑧) ≠ (𝐹𝑤) → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅))))
4746ralrn 6837 . . . . . 6 (𝐹 Fn 𝑋 → (∀𝑏 ∈ ran 𝐹((𝐹𝑧) ≠ 𝑏 → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚𝑏𝑛 ∧ (𝑚𝑛) = ∅)) ↔ ∀𝑤𝑋 ((𝐹𝑧) ≠ (𝐹𝑤) → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅))))
4847ralbidv 3191 . . . . 5 (𝐹 Fn 𝑋 → (∀𝑧𝑋𝑏 ∈ ran 𝐹((𝐹𝑧) ≠ 𝑏 → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚𝑏𝑛 ∧ (𝑚𝑛) = ∅)) ↔ ∀𝑧𝑋𝑤𝑋 ((𝐹𝑧) ≠ (𝐹𝑤) → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅))))
4941, 48bitrd 282 . . . 4 (𝐹 Fn 𝑋 → (∀𝑎 ∈ ran 𝐹𝑏 ∈ ran 𝐹(𝑎𝑏 → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)(𝑎𝑚𝑏𝑛 ∧ (𝑚𝑛) = ∅)) ↔ ∀𝑧𝑋𝑤𝑋 ((𝐹𝑧) ≠ (𝐹𝑤) → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅))))
5034, 49syl 17 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → (∀𝑎 ∈ ran 𝐹𝑏 ∈ ran 𝐹(𝑎𝑏 → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)(𝑎𝑚𝑏𝑛 ∧ (𝑚𝑛) = ∅)) ↔ ∀𝑧𝑋𝑤𝑋 ((𝐹𝑧) ≠ (𝐹𝑤) → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅))))
5132, 50mpbird 260 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → ∀𝑎 ∈ ran 𝐹𝑏 ∈ ran 𝐹(𝑎𝑏 → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)(𝑎𝑚𝑏𝑛 ∧ (𝑚𝑛) = ∅)))
521kqtopon 22323 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
5352adantr 484 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
54 ishaus2 21947 . . 3 ((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) → ((KQ‘𝐽) ∈ Haus ↔ ∀𝑎 ∈ ran 𝐹𝑏 ∈ ran 𝐹(𝑎𝑏 → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)(𝑎𝑚𝑏𝑛 ∧ (𝑚𝑛) = ∅))))
5553, 54syl 17 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → ((KQ‘𝐽) ∈ Haus ↔ ∀𝑎 ∈ ran 𝐹𝑏 ∈ ran 𝐹(𝑎𝑏 → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)(𝑎𝑚𝑏𝑛 ∧ (𝑚𝑛) = ∅))))
5651, 55mpbird 260 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ Haus)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115   ≠ wne 3013  ∀wral 3132  ∃wrex 3133  {crab 3136   ∩ cin 3917  ∅c0 4274   ↦ cmpt 5129  ran crn 5539   Fn wfn 6333  ‘cfv 6338  TopOnctopon 21506  Hauscha 21904  Regcreg 21905  KQckq 22289 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5173  ax-sep 5186  ax-nul 5193  ax-pow 5249  ax-pr 5313  ax-un 7446 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4822  df-int 4860  df-iun 4904  df-iin 4905  df-br 5050  df-opab 5112  df-mpt 5130  df-id 5443  df-xp 5544  df-rel 5545  df-cnv 5546  df-co 5547  df-dm 5548  df-rn 5549  df-res 5550  df-ima 5551  df-iota 6297  df-fun 6340  df-fn 6341  df-f 6342  df-f1 6343  df-fo 6344  df-f1o 6345  df-fv 6346  df-ov 7143  df-oprab 7144  df-mpo 7145  df-qtop 16771  df-top 21490  df-topon 21507  df-cld 21615  df-cls 21617  df-haus 21911  df-reg 21912  df-kq 22290 This theorem is referenced by:  regr1  22346
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