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Theorem hauspwpwf1 22011
Description: Lemma for hauspwpwdom 22012. 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 3982 . . . . . . . . . 10 (𝑗𝐴) ⊆ 𝐴
2 vex 3354 . . . . . . . . . . . 12 𝑗 ∈ V
32inex1 4933 . . . . . . . . . . 11 (𝑗𝐴) ∈ V
43elpw 4303 . . . . . . . . . 10 ((𝑗𝐴) ∈ 𝒫 𝐴 ↔ (𝑗𝐴) ⊆ 𝐴)
51, 4mpbir 221 . . . . . . . . 9 (𝑗𝐴) ∈ 𝒫 𝐴
6 eleq1 2838 . . . . . . . . 9 (𝑎 = (𝑗𝐴) → (𝑎 ∈ 𝒫 𝐴 ↔ (𝑗𝐴) ∈ 𝒫 𝐴))
75, 6mpbiri 248 . . . . . . . 8 (𝑎 = (𝑗𝐴) → 𝑎 ∈ 𝒫 𝐴)
87adantl 467 . . . . . . 7 ((𝑥𝑗𝑎 = (𝑗𝐴)) → 𝑎 ∈ 𝒫 𝐴)
98rexlimivw 3177 . . . . . 6 (∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴)) → 𝑎 ∈ 𝒫 𝐴)
109abssi 3826 . . . . 5 {𝑎 ∣ ∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴))} ⊆ 𝒫 𝐴
11 haustop 21356 . . . . . . . . 9 (𝐽 ∈ Haus → 𝐽 ∈ Top)
12 hauspwpwf1.x . . . . . . . . . 10 𝑋 = 𝐽
1312topopn 20931 . . . . . . . . 9 (𝐽 ∈ Top → 𝑋𝐽)
1411, 13syl 17 . . . . . . . 8 (𝐽 ∈ Haus → 𝑋𝐽)
15 ssexg 4938 . . . . . . . 8 ((𝐴𝑋𝑋𝐽) → 𝐴 ∈ V)
1614, 15sylan2 572 . . . . . . 7 ((𝐴𝑋𝐽 ∈ Haus) → 𝐴 ∈ V)
1716ancoms 455 . . . . . 6 ((𝐽 ∈ Haus ∧ 𝐴𝑋) → 𝐴 ∈ V)
18 pwexg 4981 . . . . . 6 (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
19 elpw2g 4958 . . . . . 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 248 . . . 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 750 . . . . . . . . 9 ((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) → 𝐽 ∈ Haus)
2412clsss3 21084 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋)
2511, 24sylan 561 . . . . . . . . . . 11 ((𝐽 ∈ Haus ∧ 𝐴𝑋) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋)
2625ad2antrr 697 . . . . . . . . . 10 ((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋)
27 simplrl 754 . . . . . . . . . 10 ((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) → 𝑥 ∈ ((cls‘𝐽)‘𝐴))
2826, 27sseldd 3753 . . . . . . . . 9 ((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) → 𝑥𝑋)
29 simplrr 755 . . . . . . . . . 10 ((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) → 𝑦 ∈ ((cls‘𝐽)‘𝐴))
3026, 29sseldd 3753 . . . . . . . . 9 ((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) → 𝑦𝑋)
31 simpr 471 . . . . . . . . 9 ((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) → 𝑥𝑦)
3212hausnei 21353 . . . . . . . . 9 ((𝐽 ∈ Haus ∧ (𝑥𝑋𝑦𝑋𝑥𝑦)) → ∃𝑘𝐽𝑙𝐽 (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))
3323, 28, 30, 31, 32syl13anc 1478 . . . . . . . 8 ((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) → ∃𝑘𝐽𝑙𝐽 (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))
34 simprll 756 . . . . . . . . . . . . 13 (((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) → 𝑘𝐽)
35 simprr1 1272 . . . . . . . . . . . . 13 (((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) → 𝑥𝑘)
36 eqidd 2772 . . . . . . . . . . . . 13 (((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) → (𝑘𝐴) = (𝑘𝐴))
37 elequ2 2159 . . . . . . . . . . . . . . 15 (𝑗 = 𝑘 → (𝑥𝑗𝑥𝑘))
38 ineq1 3958 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑘 → (𝑗𝐴) = (𝑘𝐴))
3938eqeq2d 2781 . . . . . . . . . . . . . . 15 (𝑗 = 𝑘 → ((𝑘𝐴) = (𝑗𝐴) ↔ (𝑘𝐴) = (𝑘𝐴)))
4037, 39anbi12d 608 . . . . . . . . . . . . . 14 (𝑗 = 𝑘 → ((𝑥𝑗 ∧ (𝑘𝐴) = (𝑗𝐴)) ↔ (𝑥𝑘 ∧ (𝑘𝐴) = (𝑘𝐴))))
4140rspcev 3460 . . . . . . . . . . . . 13 ((𝑘𝐽 ∧ (𝑥𝑘 ∧ (𝑘𝐴) = (𝑘𝐴))) → ∃𝑗𝐽 (𝑥𝑗 ∧ (𝑘𝐴) = (𝑗𝐴)))
4234, 35, 36, 41syl12anc 1474 . . . . . . . . . . . 12 (((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) → ∃𝑗𝐽 (𝑥𝑗 ∧ (𝑘𝐴) = (𝑗𝐴)))
43 vex 3354 . . . . . . . . . . . . . 14 𝑘 ∈ V
4443inex1 4933 . . . . . . . . . . . . 13 (𝑘𝐴) ∈ V
45 eqeq1 2775 . . . . . . . . . . . . . . 15 (𝑎 = (𝑘𝐴) → (𝑎 = (𝑗𝐴) ↔ (𝑘𝐴) = (𝑗𝐴)))
4645anbi2d 606 . . . . . . . . . . . . . 14 (𝑎 = (𝑘𝐴) → ((𝑥𝑗𝑎 = (𝑗𝐴)) ↔ (𝑥𝑗 ∧ (𝑘𝐴) = (𝑗𝐴))))
4746rexbidv 3200 . . . . . . . . . . . . 13 (𝑎 = (𝑘𝐴) → (∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴)) ↔ ∃𝑗𝐽 (𝑥𝑗 ∧ (𝑘𝐴) = (𝑗𝐴))))
4844, 47elab 3501 . . . . . . . . . . . 12 ((𝑘𝐴) ∈ {𝑎 ∣ ∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴))} ↔ ∃𝑗𝐽 (𝑥𝑗 ∧ (𝑘𝐴) = (𝑗𝐴)))
4942, 48sylibr 224 . . . . . . . . . . 11 (((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) → (𝑘𝐴) ∈ {𝑎 ∣ ∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴))})
5011ad2antrr 697 . . . . . . . . . . . . . . . . . . 19 (((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) → 𝐽 ∈ Top)
5150ad3antrrr 701 . . . . . . . . . . . . . . . . . 18 ((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ (𝑗𝐽𝑦𝑗)) → 𝐽 ∈ Top)
52 simplr 744 . . . . . . . . . . . . . . . . . . 19 (((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) → 𝐴𝑋)
5352ad3antrrr 701 . . . . . . . . . . . . . . . . . 18 ((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ (𝑗𝐽𝑦𝑗)) → 𝐴𝑋)
54 simprr 748 . . . . . . . . . . . . . . . . . . 19 (((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) → 𝑦 ∈ ((cls‘𝐽)‘𝐴))
5554ad3antrrr 701 . . . . . . . . . . . . . . . . . 18 ((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ (𝑗𝐽𝑦𝑗)) → 𝑦 ∈ ((cls‘𝐽)‘𝐴))
56 simplr 744 . . . . . . . . . . . . . . . . . . . 20 (((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅)) → 𝑙𝐽)
5756ad2antlr 698 . . . . . . . . . . . . . . . . . . 19 ((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ (𝑗𝐽𝑦𝑗)) → 𝑙𝐽)
58 simprl 746 . . . . . . . . . . . . . . . . . . 19 ((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ (𝑗𝐽𝑦𝑗)) → 𝑗𝐽)
59 inopn 20924 . . . . . . . . . . . . . . . . . . 19 ((𝐽 ∈ Top ∧ 𝑙𝐽𝑗𝐽) → (𝑙𝑗) ∈ 𝐽)
6051, 57, 58, 59syl3anc 1476 . . . . . . . . . . . . . . . . . 18 ((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ (𝑗𝐽𝑦𝑗)) → (𝑙𝑗) ∈ 𝐽)
61 simpr2 1235 . . . . . . . . . . . . . . . . . . . 20 (((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅)) → 𝑦𝑙)
6261ad2antlr 698 . . . . . . . . . . . . . . . . . . 19 ((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ (𝑗𝐽𝑦𝑗)) → 𝑦𝑙)
63 simprr 748 . . . . . . . . . . . . . . . . . . 19 ((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ (𝑗𝐽𝑦𝑗)) → 𝑦𝑗)
6462, 63elind 3949 . . . . . . . . . . . . . . . . . 18 ((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ (𝑗𝐽𝑦𝑗)) → 𝑦 ∈ (𝑙𝑗))
6512clsndisj 21100 . . . . . . . . . . . . . . . . . 18 (((𝐽 ∈ Top ∧ 𝐴𝑋𝑦 ∈ ((cls‘𝐽)‘𝐴)) ∧ ((𝑙𝑗) ∈ 𝐽𝑦 ∈ (𝑙𝑗))) → ((𝑙𝑗) ∩ 𝐴) ≠ ∅)
6651, 53, 55, 60, 64, 65syl32anc 1484 . . . . . . . . . . . . . . . . 17 ((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ (𝑗𝐽𝑦𝑗)) → ((𝑙𝑗) ∩ 𝐴) ≠ ∅)
67 n0 4078 . . . . . . . . . . . . . . . . 17 (((𝑙𝑗) ∩ 𝐴) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ ((𝑙𝑗) ∩ 𝐴))
6866, 67sylib 208 . . . . . . . . . . . . . . . 16 ((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ (𝑗𝐽𝑦𝑗)) → ∃𝑧 𝑧 ∈ ((𝑙𝑗) ∩ 𝐴))
69 elin 3947 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ ((𝑙𝑗) ∩ 𝐴) ↔ (𝑧 ∈ (𝑙𝑗) ∧ 𝑧𝐴))
70 elin 3947 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ (𝑙𝑗) ↔ (𝑧𝑙𝑧𝑗))
7170anbi1i 602 . . . . . . . . . . . . . . . . . . 19 ((𝑧 ∈ (𝑙𝑗) ∧ 𝑧𝐴) ↔ ((𝑧𝑙𝑧𝑗) ∧ 𝑧𝐴))
7269, 71bitri 264 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ ((𝑙𝑗) ∩ 𝐴) ↔ ((𝑧𝑙𝑧𝑗) ∧ 𝑧𝐴))
73 elin 3947 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧 ∈ (𝑗𝐴) ↔ (𝑧𝑗𝑧𝐴))
7473biimpri 218 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑧𝑗𝑧𝐴) → 𝑧 ∈ (𝑗𝐴))
7574adantll 685 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑧𝑙𝑧𝑗) ∧ 𝑧𝐴) → 𝑧 ∈ (𝑗𝐴))
7675ad2antll 700 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ ((𝑗𝐽𝑦𝑗) ∧ ((𝑧𝑙𝑧𝑗) ∧ 𝑧𝐴))) → 𝑧 ∈ (𝑗𝐴))
77 simpll 742 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑧𝑙𝑧𝑗) ∧ 𝑧𝐴) → 𝑧𝑙)
7877ad2antll 700 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ ((𝑗𝐽𝑦𝑗) ∧ ((𝑧𝑙𝑧𝑗) ∧ 𝑧𝐴))) → 𝑧𝑙)
79 simpr3 1237 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅)) → (𝑘𝑙) = ∅)
8079ad2antlr 698 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ ((𝑗𝐽𝑦𝑗) ∧ ((𝑧𝑙𝑧𝑗) ∧ 𝑧𝐴))) → (𝑘𝑙) = ∅)
81 minel 4176 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑧𝑙 ∧ (𝑘𝑙) = ∅) → ¬ 𝑧𝑘)
82 inss1 3981 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘𝐴) ⊆ 𝑘
8382sseli 3748 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 ∈ (𝑘𝐴) → 𝑧𝑘)
8481, 83nsyl 137 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑧𝑙 ∧ (𝑘𝑙) = ∅) → ¬ 𝑧 ∈ (𝑘𝐴))
8578, 80, 84syl2anc 565 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ ((𝑗𝐽𝑦𝑗) ∧ ((𝑧𝑙𝑧𝑗) ∧ 𝑧𝐴))) → ¬ 𝑧 ∈ (𝑘𝐴))
86 nelneq2 2875 . . . . . . . . . . . . . . . . . . . . 21 ((𝑧 ∈ (𝑗𝐴) ∧ ¬ 𝑧 ∈ (𝑘𝐴)) → ¬ (𝑗𝐴) = (𝑘𝐴))
8776, 85, 86syl2anc 565 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ ((𝑗𝐽𝑦𝑗) ∧ ((𝑧𝑙𝑧𝑗) ∧ 𝑧𝐴))) → ¬ (𝑗𝐴) = (𝑘𝐴))
88 eqcom 2778 . . . . . . . . . . . . . . . . . . . 20 ((𝑗𝐴) = (𝑘𝐴) ↔ (𝑘𝐴) = (𝑗𝐴))
8987, 88sylnib 317 . . . . . . . . . . . . . . . . . . 19 ((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ ((𝑗𝐽𝑦𝑗) ∧ ((𝑧𝑙𝑧𝑗) ∧ 𝑧𝐴))) → ¬ (𝑘𝐴) = (𝑗𝐴))
9089expr 444 . . . . . . . . . . . . . . . . . 18 ((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ (𝑗𝐽𝑦𝑗)) → (((𝑧𝑙𝑧𝑗) ∧ 𝑧𝐴) → ¬ (𝑘𝐴) = (𝑗𝐴)))
9172, 90syl5bi 232 . . . . . . . . . . . . . . . . 17 ((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ (𝑗𝐽𝑦𝑗)) → (𝑧 ∈ ((𝑙𝑗) ∩ 𝐴) → ¬ (𝑘𝐴) = (𝑗𝐴)))
9291exlimdv 2013 . . . . . . . . . . . . . . . 16 ((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ (𝑗𝐽𝑦𝑗)) → (∃𝑧 𝑧 ∈ ((𝑙𝑗) ∩ 𝐴) → ¬ (𝑘𝐴) = (𝑗𝐴)))
9368, 92mpd 15 . . . . . . . . . . . . . . 15 ((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ (𝑗𝐽𝑦𝑗)) → ¬ (𝑘𝐴) = (𝑗𝐴))
9493anassrs 458 . . . . . . . . . . . . . 14 (((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ 𝑗𝐽) ∧ 𝑦𝑗) → ¬ (𝑘𝐴) = (𝑗𝐴))
95 nan 814 . . . . . . . . . . . . . 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)))
9694, 95mpbir 221 . . . . . . . . . . . . 13 ((((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) ∧ 𝑗𝐽) → ¬ (𝑦𝑗 ∧ (𝑘𝐴) = (𝑗𝐴)))
9796nrexdv 3149 . . . . . . . . . . . 12 (((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) → ¬ ∃𝑗𝐽 (𝑦𝑗 ∧ (𝑘𝐴) = (𝑗𝐴)))
9845anbi2d 606 . . . . . . . . . . . . . 14 (𝑎 = (𝑘𝐴) → ((𝑦𝑗𝑎 = (𝑗𝐴)) ↔ (𝑦𝑗 ∧ (𝑘𝐴) = (𝑗𝐴))))
9998rexbidv 3200 . . . . . . . . . . . . 13 (𝑎 = (𝑘𝐴) → (∃𝑗𝐽 (𝑦𝑗𝑎 = (𝑗𝐴)) ↔ ∃𝑗𝐽 (𝑦𝑗 ∧ (𝑘𝐴) = (𝑗𝐴))))
10044, 99elab 3501 . . . . . . . . . . . 12 ((𝑘𝐴) ∈ {𝑎 ∣ ∃𝑗𝐽 (𝑦𝑗𝑎 = (𝑗𝐴))} ↔ ∃𝑗𝐽 (𝑦𝑗 ∧ (𝑘𝐴) = (𝑗𝐴)))
10197, 100sylnibr 318 . . . . . . . . . . 11 (((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) → ¬ (𝑘𝐴) ∈ {𝑎 ∣ ∃𝑗𝐽 (𝑦𝑗𝑎 = (𝑗𝐴))})
102 nelne1 3039 . . . . . . . . . . 11 (((𝑘𝐴) ∈ {𝑎 ∣ ∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴))} ∧ ¬ (𝑘𝐴) ∈ {𝑎 ∣ ∃𝑗𝐽 (𝑦𝑗𝑎 = (𝑗𝐴))}) → {𝑎 ∣ ∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴))} ≠ {𝑎 ∣ ∃𝑗𝐽 (𝑦𝑗𝑎 = (𝑗𝐴))})
10349, 101, 102syl2anc 565 . . . . . . . . . 10 (((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ ((𝑘𝐽𝑙𝐽) ∧ (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅))) → {𝑎 ∣ ∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴))} ≠ {𝑎 ∣ ∃𝑗𝐽 (𝑦𝑗𝑎 = (𝑗𝐴))})
104103expr 444 . . . . . . . . 9 (((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) ∧ (𝑘𝐽𝑙𝐽)) → ((𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅) → {𝑎 ∣ ∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴))} ≠ {𝑎 ∣ ∃𝑗𝐽 (𝑦𝑗𝑎 = (𝑗𝐴))}))
105104rexlimdvva 3186 . . . . . . . 8 ((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) → (∃𝑘𝐽𝑙𝐽 (𝑥𝑘𝑦𝑙 ∧ (𝑘𝑙) = ∅) → {𝑎 ∣ ∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴))} ≠ {𝑎 ∣ ∃𝑗𝐽 (𝑦𝑗𝑎 = (𝑗𝐴))}))
10633, 105mpd 15 . . . . . . 7 ((((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥𝑦) → {𝑎 ∣ ∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴))} ≠ {𝑎 ∣ ∃𝑗𝐽 (𝑦𝑗𝑎 = (𝑗𝐴))})
107106ex 397 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) → (𝑥𝑦 → {𝑎 ∣ ∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴))} ≠ {𝑎 ∣ ∃𝑗𝐽 (𝑦𝑗𝑎 = (𝑗𝐴))}))
108107necon4d 2967 . . . . 5 (((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) → ({𝑎 ∣ ∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴))} = {𝑎 ∣ ∃𝑗𝐽 (𝑦𝑗𝑎 = (𝑗𝐴))} → 𝑥 = 𝑦))
109 eleq1 2838 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥𝑗𝑦𝑗))
110109anbi1d 607 . . . . . . 7 (𝑥 = 𝑦 → ((𝑥𝑗𝑎 = (𝑗𝐴)) ↔ (𝑦𝑗𝑎 = (𝑗𝐴))))
111110rexbidv 3200 . . . . . 6 (𝑥 = 𝑦 → (∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴)) ↔ ∃𝑗𝐽 (𝑦𝑗𝑎 = (𝑗𝐴))))
112111abbidv 2890 . . . . 5 (𝑥 = 𝑦 → {𝑎 ∣ ∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴))} = {𝑎 ∣ ∃𝑗𝐽 (𝑦𝑗𝑎 = (𝑗𝐴))})
113108, 112impbid1 215 . . . 4 (((𝐽 ∈ Haus ∧ 𝐴𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) → ({𝑎 ∣ ∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴))} = {𝑎 ∣ ∃𝑗𝐽 (𝑦𝑗𝑎 = (𝑗𝐴))} ↔ 𝑥 = 𝑦))
114113ex 397 . . 3 ((𝐽 ∈ Haus ∧ 𝐴𝑋) → ((𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴)) → ({𝑎 ∣ ∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴))} = {𝑎 ∣ ∃𝑗𝐽 (𝑦𝑗𝑎 = (𝑗𝐴))} ↔ 𝑥 = 𝑦)))
11522, 114dom2lem 8149 . 2 ((𝐽 ∈ Haus ∧ 𝐴𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↦ {𝑎 ∣ ∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴))}):((cls‘𝐽)‘𝐴)–1-1→𝒫 𝒫 𝐴)
116 hauspwpwf1.f . . 3 𝐹 = (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↦ {𝑎 ∣ ∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴))})
117 f1eq1 6236 . . 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))
118116, 117ax-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)
119115, 118sylibr 224 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 196  wa 382  w3a 1071   = wceq 1631  wex 1852  wcel 2145  {cab 2757  wne 2943  wrex 3062  Vcvv 3351  cin 3722  wss 3723  c0 4063  𝒫 cpw 4297   cuni 4574  cmpt 4863  1-1wf1 6028  cfv 6031  Topctop 20918  clsccl 21043  Hauscha 21333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-iin 4657  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-top 20919  df-cld 21044  df-ntr 21045  df-cls 21046  df-haus 21340
This theorem is referenced by:  hauspwpwdom  22012
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