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Theorem regr1lem2 22276
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 771 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧𝑋𝑤𝑋)) ∧ (𝑎𝐽 ∧ ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅))) → 𝐽 ∈ (TopOn‘𝑋))
3 simpllr 772 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧𝑋𝑤𝑋)) ∧ (𝑎𝐽 ∧ ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅))) → 𝐽 ∈ Reg)
4 simplrl 773 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧𝑋𝑤𝑋)) ∧ (𝑎𝐽 ∧ ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅))) → 𝑧𝑋)
5 simplrr 774 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧𝑋𝑤𝑋)) ∧ (𝑎𝐽 ∧ ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅))) → 𝑤𝑋)
6 simprl 767 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧𝑋𝑤𝑋)) ∧ (𝑎𝐽 ∧ ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅))) → 𝑎𝐽)
7 simprr 769 . . . . . . . . . 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 22275 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧𝑋𝑤𝑋)) ∧ (𝑎𝐽 ∧ ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅))) → (𝑧𝑎𝑤𝑎))
9 3ancoma 1090 . . . . . . . . . . . . . 14 (((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅) ↔ ((𝐹𝑤) ∈ 𝑛 ∧ (𝐹𝑧) ∈ 𝑚 ∧ (𝑚𝑛) = ∅))
10 incom 4175 . . . . . . . . . . . . . . . 16 (𝑚𝑛) = (𝑛𝑚)
1110eqeq1i 2823 . . . . . . . . . . . . . . 15 ((𝑚𝑛) = ∅ ↔ (𝑛𝑚) = ∅)
12113anbi3i 1151 . . . . . . . . . . . . . 14 (((𝐹𝑤) ∈ 𝑛 ∧ (𝐹𝑧) ∈ 𝑚 ∧ (𝑚𝑛) = ∅) ↔ ((𝐹𝑤) ∈ 𝑛 ∧ (𝐹𝑧) ∈ 𝑚 ∧ (𝑛𝑚) = ∅))
139, 12bitri 276 . . . . . . . . . . . . 13 (((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅) ↔ ((𝐹𝑤) ∈ 𝑛 ∧ (𝐹𝑧) ∈ 𝑚 ∧ (𝑛𝑚) = ∅))
14132rexbii 3245 . . . . . . . . . . . 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 3352 . . . . . . . . . . . 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 276 . . . . . . . . . . 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 329 . . . . . . . . . 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 22275 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧𝑋𝑤𝑋)) ∧ (𝑎𝐽 ∧ ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅))) → (𝑤𝑎𝑧𝑎))
198, 18impbid 213 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧𝑋𝑤𝑋)) ∧ (𝑎𝐽 ∧ ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅))) → (𝑧𝑎𝑤𝑎))
2019expr 457 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑎𝐽) → (¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅) → (𝑧𝑎𝑤𝑎)))
2120ralrimdva 3186 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧𝑋𝑤𝑋)) → (¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅) → ∀𝑎𝐽 (𝑧𝑎𝑤𝑎)))
221kqfeq 22260 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋𝑤𝑋) → ((𝐹𝑧) = (𝐹𝑤) ↔ ∀𝑦𝐽 (𝑧𝑦𝑤𝑦)))
23 elequ2 2120 . . . . . . . . . . 11 (𝑦 = 𝑎 → (𝑧𝑦𝑧𝑎))
24 elequ2 2120 . . . . . . . . . . 11 (𝑦 = 𝑎 → (𝑤𝑦𝑤𝑎))
2523, 24bibi12d 347 . . . . . . . . . 10 (𝑦 = 𝑎 → ((𝑧𝑦𝑤𝑦) ↔ (𝑧𝑎𝑤𝑎)))
2625cbvralvw 3447 . . . . . . . . 9 (∀𝑦𝐽 (𝑧𝑦𝑤𝑦) ↔ ∀𝑎𝐽 (𝑧𝑎𝑤𝑎))
2722, 26syl6bb 288 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋𝑤𝑋) → ((𝐹𝑧) = (𝐹𝑤) ↔ ∀𝑎𝐽 (𝑧𝑎𝑤𝑎)))
28273expb 1112 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑧𝑋𝑤𝑋)) → ((𝐹𝑧) = (𝐹𝑤) ↔ ∀𝑎𝐽 (𝑧𝑎𝑤𝑎)))
2928adantlr 711 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑧𝑋𝑤𝑋)) → ((𝐹𝑧) = (𝐹𝑤) ↔ ∀𝑎𝐽 (𝑧𝑎𝑤𝑎)))
3021, 29sylibrd 260 . . . . 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 3188 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → ∀𝑧𝑋𝑤𝑋 ((𝐹𝑧) ≠ (𝐹𝑤) → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅)))
331kqffn 22261 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
3433adantr 481 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → 𝐹 Fn 𝑋)
35 neeq1 3075 . . . . . . . 8 (𝑎 = (𝐹𝑧) → (𝑎𝑏 ↔ (𝐹𝑧) ≠ 𝑏))
36 eleq1 2897 . . . . . . . . . 10 (𝑎 = (𝐹𝑧) → (𝑎𝑚 ↔ (𝐹𝑧) ∈ 𝑚))
37363anbi1d 1431 . . . . . . . . 9 (𝑎 = (𝐹𝑧) → ((𝑎𝑚𝑏𝑛 ∧ (𝑚𝑛) = ∅) ↔ ((𝐹𝑧) ∈ 𝑚𝑏𝑛 ∧ (𝑚𝑛) = ∅)))
38372rexbidv 3297 . . . . . . . 8 (𝑎 = (𝐹𝑧) → (∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)(𝑎𝑚𝑏𝑛 ∧ (𝑚𝑛) = ∅) ↔ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚𝑏𝑛 ∧ (𝑚𝑛) = ∅)))
3935, 38imbi12d 346 . . . . . . 7 (𝑎 = (𝐹𝑧) → ((𝑎𝑏 → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)(𝑎𝑚𝑏𝑛 ∧ (𝑚𝑛) = ∅)) ↔ ((𝐹𝑧) ≠ 𝑏 → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚𝑏𝑛 ∧ (𝑚𝑛) = ∅))))
4039ralbidv 3194 . . . . . 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 6846 . . . . 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 2897 . . . . . . . . . 10 (𝑏 = (𝐹𝑤) → (𝑏𝑛 ↔ (𝐹𝑤) ∈ 𝑛))
44433anbi2d 1432 . . . . . . . . 9 (𝑏 = (𝐹𝑤) → (((𝐹𝑧) ∈ 𝑚𝑏𝑛 ∧ (𝑚𝑛) = ∅) ↔ ((𝐹𝑧) ∈ 𝑚 ∧ (𝐹𝑤) ∈ 𝑛 ∧ (𝑚𝑛) = ∅)))
45442rexbidv 3297 . . . . . . . 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 346 . . . . . . 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 6846 . . . . . 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 3194 . . . . 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 280 . . . 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 258 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → ∀𝑎 ∈ ran 𝐹𝑏 ∈ ran 𝐹(𝑎𝑏 → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)(𝑎𝑚𝑏𝑛 ∧ (𝑚𝑛) = ∅)))
521kqtopon 22263 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
5352adantr 481 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
54 ishaus2 21887 . . 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 258 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ Haus)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wne 3013  wral 3135  wrex 3136  {crab 3139  cin 3932  c0 4288  cmpt 5137  ran crn 5549   Fn wfn 6343  cfv 6348  TopOnctopon 21446  Hauscha 21844  Regcreg 21845  KQckq 22229
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-qtop 16768  df-top 21430  df-topon 21447  df-cld 21555  df-cls 21557  df-haus 21851  df-reg 21852  df-kq 22230
This theorem is referenced by:  regr1  22286
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