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Theorem hauspwpwf1 22595
 Description: Lemma for hauspwpwdom 22596. Points in the closure of a set in a Hausdorff space are characterized by the open neighborhoods they extend into the generating set. (Contributed by Mario Carneiro, 28-Jul-2015.)
Hypotheses
Ref Expression
hauspwpwf1.x 𝑋 = 𝐽
hauspwpwf1.f 𝐹 = (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↦ {𝑎 ∣ ∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴))})
Assertion
Ref Expression
hauspwpwf1 ((𝐽 ∈ Haus ∧ 𝐴𝑋) → 𝐹:((cls‘𝐽)‘𝐴)–1-1→𝒫 𝒫 𝐴)
Distinct variable groups:   𝑗,𝑎,𝑥,𝐴   𝐽,𝑎,𝑗,𝑥   𝑗,𝑋,𝑥
Allowed substitution hints:   𝐹(𝑥,𝑗,𝑎)   𝑋(𝑎)

Proof of Theorem hauspwpwf1
Dummy variables 𝑘 𝑙 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inss2 4159 . . . . . . . . . 10 (𝑗𝐴) ⊆ 𝐴
2 vex 3447 . . . . . . . . . . . 12 𝑗 ∈ V
32inex1 5188 . . . . . . . . . . 11 (𝑗𝐴) ∈ V
43elpw 4504 . . . . . . . . . 10 ((𝑗𝐴) ∈ 𝒫 𝐴 ↔ (𝑗𝐴) ⊆ 𝐴)
51, 4mpbir 234 . . . . . . . . 9 (𝑗𝐴) ∈ 𝒫 𝐴
6 eleq1 2880 . . . . . . . . 9 (𝑎 = (𝑗𝐴) → (𝑎 ∈ 𝒫 𝐴 ↔ (𝑗𝐴) ∈ 𝒫 𝐴))
75, 6mpbiri 261 . . . . . . . 8 (𝑎 = (𝑗𝐴) → 𝑎 ∈ 𝒫 𝐴)
87adantl 485 . . . . . . 7 ((𝑥𝑗𝑎 = (𝑗𝐴)) → 𝑎 ∈ 𝒫 𝐴)
98rexlimivw 3244 . . . . . 6 (∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴)) → 𝑎 ∈ 𝒫 𝐴)
109abssi 4000 . . . . 5 {𝑎 ∣ ∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴))} ⊆ 𝒫 𝐴
11 haustop 21939 . . . . . . . . 9 (𝐽 ∈ Haus → 𝐽 ∈ Top)
12 hauspwpwf1.x . . . . . . . . . 10 𝑋 = 𝐽
1312topopn 21514 . . . . . . . . 9 (𝐽 ∈ Top → 𝑋𝐽)
1411, 13syl 17 . . . . . . . 8 (𝐽 ∈ Haus → 𝑋𝐽)
15 ssexg 5194 . . . . . . . 8 ((𝐴𝑋𝑋𝐽) → 𝐴 ∈ V)
1614, 15sylan2 595 . . . . . . 7 ((𝐴𝑋𝐽 ∈ Haus) → 𝐴 ∈ V)
1716ancoms 462 . . . . . 6 ((𝐽 ∈ Haus ∧ 𝐴𝑋) → 𝐴 ∈ V)
18 pwexg 5247 . . . . . 6 (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
19 elpw2g 5214 . . . . . 6 (𝒫 𝐴 ∈ V → ({𝑎 ∣ ∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴))} ∈ 𝒫 𝒫 𝐴 ↔ {𝑎 ∣ ∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴))} ⊆ 𝒫 𝐴))
2017, 18, 193syl 18 . . . . 5 ((𝐽 ∈ Haus ∧ 𝐴𝑋) → ({𝑎 ∣ ∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴))} ∈ 𝒫 𝒫 𝐴 ↔ {𝑎 ∣ ∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴))} ⊆ 𝒫 𝐴))
2110, 20mpbiri 261 . . . 4 ((𝐽 ∈ Haus ∧ 𝐴𝑋) → {𝑎 ∣ ∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴))} ∈ 𝒫 𝒫 𝐴)
2221a1d 25 . . 3 ((𝐽 ∈ Haus ∧ 𝐴𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) → {𝑎 ∣ ∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴))} ∈ 𝒫 𝒫 𝐴))
23 simplll 774 . . . . . . . . 9 ((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) → 𝐽 ∈ Haus)
2412clsss3 21667 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋)
2511, 24sylan 583 . . . . . . . . . . 11 ((𝐽 ∈ Haus ∧ 𝐴𝑋) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋)
2625ad2antrr 725 . . . . . . . . . 10 ((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋)
27 simplrl 776 . . . . . . . . . 10 ((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) → 𝑥 ∈ ((cls‘𝐽)‘𝐴))
2826, 27sseldd 3919 . . . . . . . . 9 ((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) → 𝑥𝑋)
29 simplrr 777 . . . . . . . . . 10 ((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) → 𝑦 ∈ ((cls‘𝐽)‘𝐴))
3026, 29sseldd 3919 . . . . . . . . 9 ((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) → 𝑦𝑋)
31 simpr 488 . . . . . . . . 9 ((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) → 𝑥𝑦)
3212hausnei 21936 . . . . . . . . 9 ((𝐽 ∈ Haus ∧ (𝑥𝑋𝑦𝑋𝑥𝑦)) → ∃𝑘𝐽𝑙𝐽 (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))
3323, 28, 30, 31, 32syl13anc 1369 . . . . . . . 8 ((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) → ∃𝑘𝐽𝑙𝐽 (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))
34 simprll 778 . . . . . . . . . . . . 13 (((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) → 𝑘𝐽)
35 simprr1 1218 . . . . . . . . . . . . 13 (((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) → 𝑥𝑘)
36 eqidd 2802 . . . . . . . . . . . . 13 (((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) → (𝑘𝐴) = (𝑘𝐴))
37 elequ2 2127 . . . . . . . . . . . . . . 15 (𝑗 = 𝑘 → (𝑥𝑗𝑥𝑘))
38 ineq1 4134 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑘 → (𝑗𝐴) = (𝑘𝐴))
3938eqeq2d 2812 . . . . . . . . . . . . . . 15 (𝑗 = 𝑘 → ((𝑘𝐴) = (𝑗𝐴) ↔ (𝑘𝐴) = (𝑘𝐴)))
4037, 39anbi12d 633 . . . . . . . . . . . . . 14 (𝑗 = 𝑘 → ((𝑥𝑗 ∧ (𝑘𝐴) = (𝑗𝐴)) ↔ (𝑥𝑘 ∧ (𝑘𝐴) = (𝑘𝐴))))
4140rspcev 3574 . . . . . . . . . . . . 13 ((𝑘𝐽 ∧ (𝑥𝑘 ∧ (𝑘𝐴) = (𝑘𝐴))) → ∃𝑗𝐽 (𝑥𝑗 ∧ (𝑘𝐴) = (𝑗𝐴)))
4234, 35, 36, 41syl12anc 835 . . . . . . . . . . . 12 (((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) → ∃𝑗𝐽 (𝑥𝑗 ∧ (𝑘𝐴) = (𝑗𝐴)))
43 vex 3447 . . . . . . . . . . . . . 14 𝑘 ∈ V
4443inex1 5188 . . . . . . . . . . . . 13 (𝑘𝐴) ∈ V
45 eqeq1 2805 . . . . . . . . . . . . . . 15 (𝑎 = (𝑘𝐴) → (𝑎 = (𝑗𝐴) ↔ (𝑘𝐴) = (𝑗𝐴)))
4645anbi2d 631 . . . . . . . . . . . . . 14 (𝑎 = (𝑘𝐴) → ((𝑥𝑗𝑎 = (𝑗𝐴)) ↔ (𝑥𝑗 ∧ (𝑘𝐴) = (𝑗𝐴))))
4746rexbidv 3259 . . . . . . . . . . . . 13 (𝑎 = (𝑘𝐴) → (∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴)) ↔ ∃𝑗𝐽 (𝑥𝑗 ∧ (𝑘𝐴) = (𝑗𝐴))))
4844, 47elab 3618 . . . . . . . . . . . 12 ((𝑘𝐴) ∈ {𝑎 ∣ ∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴))} ↔ ∃𝑗𝐽 (𝑥𝑗 ∧ (𝑘𝐴) = (𝑗𝐴)))
4942, 48sylibr 237 . . . . . . . . . . 11 (((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) → (𝑘𝐴) ∈ {𝑎 ∣ ∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴))})
5011ad2antrr 725 . . . . . . . . . . . . . . . . . . 19 (((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) → 𝐽 ∈ Top)
5150ad3antrrr 729 . . . . . . . . . . . . . . . . . 18 ((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ (𝑗𝐽𝑦𝑗)) → 𝐽 ∈ Top)
52 simplr 768 . . . . . . . . . . . . . . . . . . 19 (((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) → 𝐴𝑋)
5352ad3antrrr 729 . . . . . . . . . . . . . . . . . 18 ((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ (𝑗𝐽𝑦𝑗)) → 𝐴𝑋)
54 simprr 772 . . . . . . . . . . . . . . . . . . 19 (((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) → 𝑦 ∈ ((cls‘𝐽)‘𝐴))
5554ad3antrrr 729 . . . . . . . . . . . . . . . . . 18 ((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ (𝑗𝐽𝑦𝑗)) → 𝑦 ∈ ((cls‘𝐽)‘𝐴))
56 simplr 768 . . . . . . . . . . . . . . . . . . . 20 (((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅)) → 𝑙𝐽)
5756ad2antlr 726 . . . . . . . . . . . . . . . . . . 19 ((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ (𝑗𝐽𝑦𝑗)) → 𝑙𝐽)
58 simprl 770 . . . . . . . . . . . . . . . . . . 19 ((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ (𝑗𝐽𝑦𝑗)) → 𝑗𝐽)
59 inopn 21507 . . . . . . . . . . . . . . . . . . 19 ((𝐽 ∈ Top ∧ 𝑙𝐽𝑗𝐽) → (𝑙𝑗) ∈ 𝐽)
6051, 57, 58, 59syl3anc 1368 . . . . . . . . . . . . . . . . . 18 ((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ (𝑗𝐽𝑦𝑗)) → (𝑙𝑗) ∈ 𝐽)
61 simpr2 1192 . . . . . . . . . . . . . . . . . . . 20 (((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅)) → 𝑦𝑙)
6261ad2antlr 726 . . . . . . . . . . . . . . . . . . 19 ((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ (𝑗𝐽𝑦𝑗)) → 𝑦𝑙)
63 simprr 772 . . . . . . . . . . . . . . . . . . 19 ((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ (𝑗𝐽𝑦𝑗)) → 𝑦𝑗)
6462, 63elind 4124 . . . . . . . . . . . . . . . . . 18 ((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ (𝑗𝐽𝑦𝑗)) → 𝑦 ∈ (𝑙𝑗))
6512clsndisj 21683 . . . . . . . . . . . . . . . . . 18 (((𝐽 ∈ Top ∧ 𝐴𝑋𝑦 ∈ ((cls‘𝐽)‘𝐴)) ∧ ((𝑙𝑗) ∈ 𝐽𝑦 ∈ (𝑙𝑗))) → ((𝑙𝑗) ∩ 𝐴) ≠ ∅)
6651, 53, 55, 60, 64, 65syl32anc 1375 . . . . . . . . . . . . . . . . 17 ((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ (𝑗𝐽𝑦𝑗)) → ((𝑙𝑗) ∩ 𝐴) ≠ ∅)
67 n0 4263 . . . . . . . . . . . . . . . . 17 (((𝑙𝑗) ∩ 𝐴) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ ((𝑙𝑗) ∩ 𝐴))
6866, 67sylib 221 . . . . . . . . . . . . . . . 16 ((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ (𝑗𝐽𝑦𝑗)) → ∃𝑧 𝑧 ∈ ((𝑙𝑗) ∩ 𝐴))
69 elin 3900 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ ((𝑙𝑗) ∩ 𝐴) ↔ (𝑧 ∈ (𝑙𝑗) ∧ 𝑧𝐴))
70 elin 3900 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ (𝑙𝑗) ↔ (𝑧𝑙𝑧𝑗))
7170anbi1i 626 . . . . . . . . . . . . . . . . . . 19 ((𝑧 ∈ (𝑙𝑗) ∧ 𝑧𝐴) ↔ ((𝑧𝑙𝑧𝑗) ∧ 𝑧𝐴))
7269, 71bitri 278 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ ((𝑙𝑗) ∩ 𝐴) ↔ ((𝑧𝑙𝑧𝑗) ∧ 𝑧𝐴))
73 elin 3900 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧 ∈ (𝑗𝐴) ↔ (𝑧𝑗𝑧𝐴))
7473biimpri 231 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑧𝑗𝑧𝐴) → 𝑧 ∈ (𝑗𝐴))
7574adantll 713 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑧𝑙𝑧𝑗) ∧ 𝑧𝐴) → 𝑧 ∈ (𝑗𝐴))
7675ad2antll 728 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ ((𝑗𝐽𝑦𝑗) ∧ ((𝑧𝑙𝑧𝑗) ∧ 𝑧𝐴))) → 𝑧 ∈ (𝑗𝐴))
77 simpll 766 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑧𝑙𝑧𝑗) ∧ 𝑧𝐴) → 𝑧𝑙)
7877ad2antll 728 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ ((𝑗𝐽𝑦𝑗) ∧ ((𝑧𝑙𝑧𝑗) ∧ 𝑧𝐴))) → 𝑧𝑙)
79 simpr3 1193 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅)) → (𝑘𝑙) = ∅)
8079ad2antlr 726 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ ((𝑗𝐽𝑦𝑗) ∧ ((𝑧𝑙𝑧𝑗) ∧ 𝑧𝐴))) → (𝑘𝑙) = ∅)
81 minel 4376 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑧𝑙 ∧ (𝑘𝑙) = ∅) → ¬ 𝑧𝑘)
82 elinel1 4125 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 ∈ (𝑘𝐴) → 𝑧𝑘)
8381, 82nsyl 142 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑧𝑙 ∧ (𝑘𝑙) = ∅) → ¬ 𝑧 ∈ (𝑘𝐴))
8478, 80, 83syl2anc 587 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ ((𝑗𝐽𝑦𝑗) ∧ ((𝑧𝑙𝑧𝑗) ∧ 𝑧𝐴))) → ¬ 𝑧 ∈ (𝑘𝐴))
85 nelneq2 2918 . . . . . . . . . . . . . . . . . . . . 21 ((𝑧 ∈ (𝑗𝐴) ∧ ¬ 𝑧 ∈ (𝑘𝐴)) → ¬ (𝑗𝐴) = (𝑘𝐴))
8676, 84, 85syl2anc 587 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ ((𝑗𝐽𝑦𝑗) ∧ ((𝑧𝑙𝑧𝑗) ∧ 𝑧𝐴))) → ¬ (𝑗𝐴) = (𝑘𝐴))
87 eqcom 2808 . . . . . . . . . . . . . . . . . . . 20 ((𝑗𝐴) = (𝑘𝐴) ↔ (𝑘𝐴) = (𝑗𝐴))
8886, 87sylnib 331 . . . . . . . . . . . . . . . . . . 19 ((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ ((𝑗𝐽𝑦𝑗) ∧ ((𝑧𝑙𝑧𝑗) ∧ 𝑧𝐴))) → ¬ (𝑘𝐴) = (𝑗𝐴))
8988expr 460 . . . . . . . . . . . . . . . . . 18 ((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ (𝑗𝐽𝑦𝑗)) → (((𝑧𝑙𝑧𝑗) ∧ 𝑧𝐴) → ¬ (𝑘𝐴) = (𝑗𝐴)))
9072, 89syl5bi 245 . . . . . . . . . . . . . . . . 17 ((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ (𝑗𝐽𝑦𝑗)) → (𝑧 ∈ ((𝑙𝑗) ∩ 𝐴) → ¬ (𝑘𝐴) = (𝑗𝐴)))
9190exlimdv 1934 . . . . . . . . . . . . . . . 16 ((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ (𝑗𝐽𝑦𝑗)) → (∃𝑧 𝑧 ∈ ((𝑙𝑗) ∩ 𝐴) → ¬ (𝑘𝐴) = (𝑗𝐴)))
9268, 91mpd 15 . . . . . . . . . . . . . . 15 ((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ (𝑗𝐽𝑦𝑗)) → ¬ (𝑘𝐴) = (𝑗𝐴))
9392anassrs 471 . . . . . . . . . . . . . 14 (((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ 𝑗𝐽) ∧ 𝑦𝑗) → ¬ (𝑘𝐴) = (𝑗𝐴))
94 nan 828 . . . . . . . . . . . . . 14 (((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ 𝑗𝐽) → ¬ (𝑦𝑗 ∧ (𝑘𝐴) = (𝑗𝐴))) ↔ (((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ 𝑗𝐽) ∧ 𝑦𝑗) → ¬ (𝑘𝐴) = (𝑗𝐴)))
9593, 94mpbir 234 . . . . . . . . . . . . 13 ((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ 𝑗𝐽) → ¬ (𝑦𝑗 ∧ (𝑘𝐴) = (𝑗𝐴)))
9695nrexdv 3232 . . . . . . . . . . . 12 (((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) → ¬ ∃𝑗𝐽 (𝑦𝑗 ∧ (𝑘𝐴) = (𝑗𝐴)))
9745anbi2d 631 . . . . . . . . . . . . . 14 (𝑎 = (𝑘𝐴) → ((𝑦𝑗𝑎 = (𝑗𝐴)) ↔ (𝑦𝑗 ∧ (𝑘𝐴) = (𝑗𝐴))))
9897rexbidv 3259 . . . . . . . . . . . . 13 (𝑎 = (𝑘𝐴) → (∃𝑗𝐽 (𝑦𝑗𝑎 = (𝑗𝐴)) ↔ ∃𝑗𝐽 (𝑦𝑗 ∧ (𝑘𝐴) = (𝑗𝐴))))
9944, 98elab 3618 . . . . . . . . . . . 12 ((𝑘𝐴) ∈ {𝑎 ∣ ∃𝑗𝐽 (𝑦𝑗𝑎 = (𝑗𝐴))} ↔ ∃𝑗𝐽 (𝑦𝑗 ∧ (𝑘𝐴) = (𝑗𝐴)))
10096, 99sylnibr 332 . . . . . . . . . . 11 (((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) → ¬ (𝑘𝐴) ∈ {𝑎 ∣ ∃𝑗𝐽 (𝑦𝑗𝑎 = (𝑗𝐴))})
101 nelne1 3086 . . . . . . . . . . 11 (((𝑘𝐴) ∈ {𝑎 ∣ ∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴))} ∧ ¬ (𝑘𝐴) ∈ {𝑎 ∣ ∃𝑗𝐽 (𝑦𝑗𝑎 = (𝑗𝐴))}) → {𝑎 ∣ ∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴))} ≠ {𝑎 ∣ ∃𝑗𝐽 (𝑦𝑗𝑎 = (𝑗𝐴))})
10249, 100, 101syl2anc 587 . . . . . . . . . 10 (((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) → {𝑎 ∣ ∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴))} ≠ {𝑎 ∣ ∃𝑗𝐽 (𝑦𝑗𝑎 = (𝑗𝐴))})
103102expr 460 . . . . . . . . 9 (((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ (𝑘𝐽𝑙𝐽)) → ((𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅) → {𝑎 ∣ ∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴))} ≠ {𝑎 ∣ ∃𝑗𝐽 (𝑦𝑗𝑎 = (𝑗𝐴))}))
104103rexlimdvva 3256 . . . . . . . 8 ((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) → (∃𝑘𝐽𝑙𝐽 (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅) → {𝑎 ∣ ∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴))} ≠ {𝑎 ∣ ∃𝑗𝐽 (𝑦𝑗𝑎 = (𝑗𝐴))}))
10533, 104mpd 15 . . . . . . 7 ((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) → {𝑎 ∣ ∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴))} ≠ {𝑎 ∣ ∃𝑗𝐽 (𝑦𝑗𝑎 = (𝑗𝐴))})
106105ex 416 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) → (𝑥𝑦 → {𝑎 ∣ ∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴))} ≠ {𝑎 ∣ ∃𝑗𝐽 (𝑦𝑗𝑎 = (𝑗𝐴))}))
107106necon4d 3014 . . . . 5 (((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) → ({𝑎 ∣ ∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴))} = {𝑎 ∣ ∃𝑗𝐽 (𝑦𝑗𝑎 = (𝑗𝐴))} → 𝑥 = 𝑦))
108 eleq1 2880 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥𝑗𝑦𝑗))
109108anbi1d 632 . . . . . . 7 (𝑥 = 𝑦 → ((𝑥𝑗𝑎 = (𝑗𝐴)) ↔ (𝑦𝑗𝑎 = (𝑗𝐴))))
110109rexbidv 3259 . . . . . 6 (𝑥 = 𝑦 → (∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴)) ↔ ∃𝑗𝐽 (𝑦𝑗𝑎 = (𝑗𝐴))))
111110abbidv 2865 . . . . 5 (𝑥 = 𝑦 → {𝑎 ∣ ∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴))} = {𝑎 ∣ ∃𝑗𝐽 (𝑦𝑗𝑎 = (𝑗𝐴))})
112107, 111impbid1 228 . . . 4 (((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) → ({𝑎 ∣ ∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴))} = {𝑎 ∣ ∃𝑗𝐽 (𝑦𝑗𝑎 = (𝑗𝐴))} ↔ 𝑥 = 𝑦))
113112ex 416 . . 3 ((𝐽 ∈ Haus ∧ 𝐴𝑋) → ((𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴)) → ({𝑎 ∣ ∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴))} = {𝑎 ∣ ∃𝑗𝐽 (𝑦𝑗𝑎 = (𝑗𝐴))} ↔ 𝑥 = 𝑦)))
11422, 113dom2lem 8536 . 2 ((𝐽 ∈ Haus ∧ 𝐴𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↦ {𝑎 ∣ ∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴))}):((cls‘𝐽)‘𝐴)–1-1→𝒫 𝒫 𝐴)
115 hauspwpwf1.f . . 3 𝐹 = (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↦ {𝑎 ∣ ∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴))})
116 f1eq1 6548 . . 3 (𝐹 = (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↦ {𝑎 ∣ ∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴))}) → (𝐹:((cls‘𝐽)‘𝐴)–1-1→𝒫 𝒫 𝐴 ↔ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↦ {𝑎 ∣ ∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴))}):((cls‘𝐽)‘𝐴)–1-1→𝒫 𝒫 𝐴))
117115, 116ax-mp 5 . 2 (𝐹:((cls‘𝐽)‘𝐴)–1-1→𝒫 𝒫 𝐴 ↔ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↦ {𝑎 ∣ ∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴))}):((cls‘𝐽)‘𝐴)–1-1→𝒫 𝒫 𝐴)
118114, 117sylibr 237 1 ((𝐽 ∈ Haus ∧ 𝐴𝑋) → 𝐹:((cls‘𝐽)‘𝐴)–1-1→𝒫 𝒫 𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2112  {cab 2779   ≠ wne 2990  ∃wrex 3110  Vcvv 3444   ∩ cin 3883   ⊆ wss 3884  ∅c0 4246  𝒫 cpw 4500  ∪ cuni 4803   ↦ cmpt 5113  –1-1→wf1 6325  ‘cfv 6328  Topctop 21501  clsccl 21626  Hauscha 21916 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 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 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-top 21502  df-cld 21627  df-ntr 21628  df-cls 21629  df-haus 21923 This theorem is referenced by:  hauspwpwdom  22596
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