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Theorem regr1lem2 21837
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 791 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧𝑋𝑤𝑋)) ∧ (𝑎𝐽 ∧ ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅))) → 𝐽 ∈ (TopOn‘𝑋))
3 simpllr 793 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧𝑋𝑤𝑋)) ∧ (𝑎𝐽 ∧ ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅))) → 𝐽 ∈ Reg)
4 simplrl 795 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧𝑋𝑤𝑋)) ∧ (𝑎𝐽 ∧ ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅))) → 𝑧𝑋)
5 simplrr 796 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧𝑋𝑤𝑋)) ∧ (𝑎𝐽 ∧ ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅))) → 𝑤𝑋)
6 simprl 787 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧𝑋𝑤𝑋)) ∧ (𝑎𝐽 ∧ ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅))) → 𝑎𝐽)
7 simprr 789 . . . . . . . . . 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 21836 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧𝑋𝑤𝑋)) ∧ (𝑎𝐽 ∧ ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅))) → (𝑧𝑎𝑤𝑎))
9 3ancoma 1119 . . . . . . . . . . . . . 14 (((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅) ↔ ((𝐹𝑤) ∈ 𝑛 ∧ (𝐹𝑧) ∈ 𝑚 ∧ (𝑚𝑛) = ∅))
10 incom 3969 . . . . . . . . . . . . . . . 16 (𝑚𝑛) = (𝑛𝑚)
1110eqeq1i 2770 . . . . . . . . . . . . . . 15 ((𝑚𝑛) = ∅ ↔ (𝑛𝑚) = ∅)
12113anbi3i 1198 . . . . . . . . . . . . . 14 (((𝐹𝑤) ∈ 𝑛 ∧ (𝐹𝑧) ∈ 𝑚 ∧ (𝑚𝑛) = ∅) ↔ ((𝐹𝑤) ∈ 𝑛 ∧ (𝐹𝑧) ∈ 𝑚 ∧ (𝑛𝑚) = ∅))
139, 12bitri 266 . . . . . . . . . . . . 13 (((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅) ↔ ((𝐹𝑤) ∈ 𝑛 ∧ (𝐹𝑧) ∈ 𝑚 ∧ (𝑛𝑚) = ∅))
14132rexbii 3189 . . . . . . . . . . . 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 3246 . . . . . . . . . . . 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 266 . . . . . . . . . . 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 319 . . . . . . . . . 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 21836 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧𝑋𝑤𝑋)) ∧ (𝑎𝐽 ∧ ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅))) → (𝑤𝑎𝑧𝑎))
198, 18impbid 203 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧𝑋𝑤𝑋)) ∧ (𝑎𝐽 ∧ ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅))) → (𝑧𝑎𝑤𝑎))
2019expr 448 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑎𝐽) → (¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅) → (𝑧𝑎𝑤𝑎)))
2120ralrimdva 3116 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧𝑋𝑤𝑋)) → (¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅) → ∀𝑎𝐽 (𝑧𝑎𝑤𝑎)))
221kqfeq 21821 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋𝑤𝑋) → ((𝐹𝑧) = (𝐹𝑤) ↔ ∀𝑦𝐽 (𝑧𝑦𝑤𝑦)))
23 elequ2 2169 . . . . . . . . . . 11 (𝑦 = 𝑎 → (𝑧𝑦𝑧𝑎))
24 elequ2 2169 . . . . . . . . . . 11 (𝑦 = 𝑎 → (𝑤𝑦𝑤𝑎))
2523, 24bibi12d 336 . . . . . . . . . 10 (𝑦 = 𝑎 → ((𝑧𝑦𝑤𝑦) ↔ (𝑧𝑎𝑤𝑎)))
2625cbvralv 3319 . . . . . . . . 9 (∀𝑦𝐽 (𝑧𝑦𝑤𝑦) ↔ ∀𝑎𝐽 (𝑧𝑎𝑤𝑎))
2722, 26syl6bb 278 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋𝑤𝑋) → ((𝐹𝑧) = (𝐹𝑤) ↔ ∀𝑎𝐽 (𝑧𝑎𝑤𝑎)))
28273expb 1149 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑧𝑋𝑤𝑋)) → ((𝐹𝑧) = (𝐹𝑤) ↔ ∀𝑎𝐽 (𝑧𝑎𝑤𝑎)))
2928adantlr 706 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧𝑋𝑤𝑋)) → ((𝐹𝑧) = (𝐹𝑤) ↔ ∀𝑎𝐽 (𝑧𝑎𝑤𝑎)))
3021, 29sylibrd 250 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧𝑋𝑤𝑋)) → (¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅) → (𝐹𝑧) = (𝐹𝑤)))
3130necon1ad 2954 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧𝑋𝑤𝑋)) → ((𝐹𝑧) ≠ (𝐹𝑤) → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅)))
3231ralrimivva 3118 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → ∀𝑧𝑋𝑤𝑋 ((𝐹𝑧) ≠ (𝐹𝑤) → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅)))
331kqffn 21822 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
3433adantr 472 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → 𝐹 Fn 𝑋)
35 neeq1 2999 . . . . . . . 8 (𝑎 = (𝐹𝑧) → (𝑎𝑏 ↔ (𝐹𝑧) ≠ 𝑏))
36 eleq1 2832 . . . . . . . . . 10 (𝑎 = (𝐹𝑧) → (𝑎𝑚 ↔ (𝐹𝑧) ∈ 𝑚))
37363anbi1d 1564 . . . . . . . . 9 (𝑎 = (𝐹𝑧) → ((𝑎𝑚𝑏𝑛 ∧ (𝑚𝑛) = ∅) ↔ ((𝐹𝑧) ∈ 𝑚𝑏𝑛 ∧ (𝑚𝑛) = ∅)))
38372rexbidv 3204 . . . . . . . 8 (𝑎 = (𝐹𝑧) → (∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)(𝑎𝑚𝑏𝑛 ∧ (𝑚𝑛) = ∅) ↔ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚𝑏𝑛 ∧ (𝑚𝑛) = ∅)))
3935, 38imbi12d 335 . . . . . . 7 (𝑎 = (𝐹𝑧) → ((𝑎𝑏 → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)(𝑎𝑚𝑏𝑛 ∧ (𝑚𝑛) = ∅)) ↔ ((𝐹𝑧) ≠ 𝑏 → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚𝑏𝑛 ∧ (𝑚𝑛) = ∅))))
4039ralbidv 3133 . . . . . 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 6556 . . . . 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 3000 . . . . . . . 8 (𝑏 = (𝐹𝑤) → ((𝐹𝑧) ≠ 𝑏 ↔ (𝐹𝑧) ≠ (𝐹𝑤)))
43 eleq1 2832 . . . . . . . . . 10 (𝑏 = (𝐹𝑤) → (𝑏𝑛 ↔ (𝐹𝑤) ∈ 𝑛))
44433anbi2d 1565 . . . . . . . . 9 (𝑏 = (𝐹𝑤) → (((𝐹𝑧) ∈ 𝑚𝑏𝑛 ∧ (𝑚𝑛) = ∅) ↔ ((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅)))
45442rexbidv 3204 . . . . . . . 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 335 . . . . . . 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 6556 . . . . . 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 3133 . . . . 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 270 . . . 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 248 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → ∀𝑎 ∈ ran 𝐹𝑏 ∈ ran 𝐹(𝑎𝑏 → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)(𝑎𝑚𝑏𝑛 ∧ (𝑚𝑛) = ∅)))
521kqtopon 21824 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
5352adantr 472 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
54 ishaus2 21449 . . 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 248 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ Haus)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wne 2937  wral 3055  wrex 3056  {crab 3059  cin 3733  c0 4081  cmpt 4890  ran crn 5280   Fn wfn 6065  cfv 6070  TopOnctopon 21008  Hauscha 21406  Regcreg 21407  KQckq 21790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-iin 4681  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-qtop 16447  df-top 20992  df-topon 21009  df-cld 21117  df-cls 21119  df-haus 21413  df-reg 21414  df-kq 21791
This theorem is referenced by:  regr1  21847
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