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Theorem hauscmplem 21411
Description: Lemma for hauscmp 21412. (Contributed by Mario Carneiro, 27-Nov-2013.)
Hypotheses
Ref Expression
hauscmp.1 𝑋 = 𝐽
hauscmplem.2 𝑂 = {𝑦𝐽 ∣ ∃𝑤𝐽 (𝐴𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑋𝑦))}
hauscmplem.3 (𝜑𝐽 ∈ Haus)
hauscmplem.4 (𝜑𝑆𝑋)
hauscmplem.5 (𝜑 → (𝐽t 𝑆) ∈ Comp)
hauscmplem.6 (𝜑𝐴 ∈ (𝑋𝑆))
Assertion
Ref Expression
hauscmplem (𝜑 → ∃𝑧𝐽 (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ (𝑋𝑆)))
Distinct variable groups:   𝑦,𝑤,𝑧,𝐴   𝑤,𝐽,𝑦,𝑧   𝜑,𝑤,𝑦,𝑧   𝑤,𝑆,𝑦,𝑧   𝑧,𝑂   𝑤,𝑋,𝑦,𝑧
Allowed substitution hints:   𝑂(𝑦,𝑤)

Proof of Theorem hauscmplem
Dummy variables 𝑓 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hauscmplem.3 . . . . . . 7 (𝜑𝐽 ∈ Haus)
2 haustop 21337 . . . . . . 7 (𝐽 ∈ Haus → 𝐽 ∈ Top)
31, 2syl 17 . . . . . 6 (𝜑𝐽 ∈ Top)
43ad3antrrr 768 . . . . 5 ((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ 𝑆 𝑥) ∧ 𝑥 = ∅) → 𝐽 ∈ Top)
5 hauscmp.1 . . . . . 6 𝑋 = 𝐽
65topopn 20913 . . . . 5 (𝐽 ∈ Top → 𝑋𝐽)
74, 6syl 17 . . . 4 ((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ 𝑆 𝑥) ∧ 𝑥 = ∅) → 𝑋𝐽)
8 hauscmplem.6 . . . . . 6 (𝜑𝐴 ∈ (𝑋𝑆))
98eldifad 3727 . . . . 5 (𝜑𝐴𝑋)
109ad3antrrr 768 . . . 4 ((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ 𝑆 𝑥) ∧ 𝑥 = ∅) → 𝐴𝑋)
115clstop 21075 . . . . . . 7 (𝐽 ∈ Top → ((cls‘𝐽)‘𝑋) = 𝑋)
124, 11syl 17 . . . . . 6 ((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ 𝑆 𝑥) ∧ 𝑥 = ∅) → ((cls‘𝐽)‘𝑋) = 𝑋)
13 simplr 809 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ 𝑆 𝑥) ∧ 𝑥 = ∅) → 𝑆 𝑥)
14 unieq 4596 . . . . . . . . . . . 12 (𝑥 = ∅ → 𝑥 = ∅)
15 uni0 4617 . . . . . . . . . . . 12 ∅ = ∅
1614, 15syl6eq 2810 . . . . . . . . . . 11 (𝑥 = ∅ → 𝑥 = ∅)
1716adantl 473 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ 𝑆 𝑥) ∧ 𝑥 = ∅) → 𝑥 = ∅)
1813, 17sseqtrd 3782 . . . . . . . . 9 ((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ 𝑆 𝑥) ∧ 𝑥 = ∅) → 𝑆 ⊆ ∅)
19 ss0 4117 . . . . . . . . 9 (𝑆 ⊆ ∅ → 𝑆 = ∅)
2018, 19syl 17 . . . . . . . 8 ((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ 𝑆 𝑥) ∧ 𝑥 = ∅) → 𝑆 = ∅)
2120difeq2d 3871 . . . . . . 7 ((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ 𝑆 𝑥) ∧ 𝑥 = ∅) → (𝑋𝑆) = (𝑋 ∖ ∅))
22 dif0 4093 . . . . . . 7 (𝑋 ∖ ∅) = 𝑋
2321, 22syl6eq 2810 . . . . . 6 ((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ 𝑆 𝑥) ∧ 𝑥 = ∅) → (𝑋𝑆) = 𝑋)
2412, 23eqtr4d 2797 . . . . 5 ((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ 𝑆 𝑥) ∧ 𝑥 = ∅) → ((cls‘𝐽)‘𝑋) = (𝑋𝑆))
25 eqimss 3798 . . . . 5 (((cls‘𝐽)‘𝑋) = (𝑋𝑆) → ((cls‘𝐽)‘𝑋) ⊆ (𝑋𝑆))
2624, 25syl 17 . . . 4 ((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ 𝑆 𝑥) ∧ 𝑥 = ∅) → ((cls‘𝐽)‘𝑋) ⊆ (𝑋𝑆))
27 eleq2 2828 . . . . . 6 (𝑧 = 𝑋 → (𝐴𝑧𝐴𝑋))
28 fveq2 6352 . . . . . . 7 (𝑧 = 𝑋 → ((cls‘𝐽)‘𝑧) = ((cls‘𝐽)‘𝑋))
2928sseq1d 3773 . . . . . 6 (𝑧 = 𝑋 → (((cls‘𝐽)‘𝑧) ⊆ (𝑋𝑆) ↔ ((cls‘𝐽)‘𝑋) ⊆ (𝑋𝑆)))
3027, 29anbi12d 749 . . . . 5 (𝑧 = 𝑋 → ((𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ (𝑋𝑆)) ↔ (𝐴𝑋 ∧ ((cls‘𝐽)‘𝑋) ⊆ (𝑋𝑆))))
3130rspcev 3449 . . . 4 ((𝑋𝐽 ∧ (𝐴𝑋 ∧ ((cls‘𝐽)‘𝑋) ⊆ (𝑋𝑆))) → ∃𝑧𝐽 (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ (𝑋𝑆)))
327, 10, 26, 31syl12anc 1475 . . 3 ((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ 𝑆 𝑥) ∧ 𝑥 = ∅) → ∃𝑧𝐽 (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ (𝑋𝑆)))
33 elin 3939 . . . . . . 7 (𝑥 ∈ (𝒫 𝑂 ∩ Fin) ↔ (𝑥 ∈ 𝒫 𝑂𝑥 ∈ Fin))
34 id 22 . . . . . . . 8 (𝑥 ∈ Fin → 𝑥 ∈ Fin)
35 elpwi 4312 . . . . . . . . . . 11 (𝑥 ∈ 𝒫 𝑂𝑥𝑂)
3635sseld 3743 . . . . . . . . . 10 (𝑥 ∈ 𝒫 𝑂 → (𝑧𝑥𝑧𝑂))
37 difeq2 3865 . . . . . . . . . . . . . . 15 (𝑦 = 𝑧 → (𝑋𝑦) = (𝑋𝑧))
3837sseq2d 3774 . . . . . . . . . . . . . 14 (𝑦 = 𝑧 → (((cls‘𝐽)‘𝑤) ⊆ (𝑋𝑦) ↔ ((cls‘𝐽)‘𝑤) ⊆ (𝑋𝑧)))
3938anbi2d 742 . . . . . . . . . . . . 13 (𝑦 = 𝑧 → ((𝐴𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑋𝑦)) ↔ (𝐴𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑋𝑧))))
4039rexbidv 3190 . . . . . . . . . . . 12 (𝑦 = 𝑧 → (∃𝑤𝐽 (𝐴𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑋𝑦)) ↔ ∃𝑤𝐽 (𝐴𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑋𝑧))))
41 hauscmplem.2 . . . . . . . . . . . 12 𝑂 = {𝑦𝐽 ∣ ∃𝑤𝐽 (𝐴𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑋𝑦))}
4240, 41elrab2 3507 . . . . . . . . . . 11 (𝑧𝑂 ↔ (𝑧𝐽 ∧ ∃𝑤𝐽 (𝐴𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑋𝑧))))
4342simprbi 483 . . . . . . . . . 10 (𝑧𝑂 → ∃𝑤𝐽 (𝐴𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑋𝑧)))
4436, 43syl6 35 . . . . . . . . 9 (𝑥 ∈ 𝒫 𝑂 → (𝑧𝑥 → ∃𝑤𝐽 (𝐴𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑋𝑧))))
4544ralrimiv 3103 . . . . . . . 8 (𝑥 ∈ 𝒫 𝑂 → ∀𝑧𝑥𝑤𝐽 (𝐴𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑋𝑧)))
46 eleq2 2828 . . . . . . . . . 10 (𝑤 = (𝑓𝑧) → (𝐴𝑤𝐴 ∈ (𝑓𝑧)))
47 fveq2 6352 . . . . . . . . . . 11 (𝑤 = (𝑓𝑧) → ((cls‘𝐽)‘𝑤) = ((cls‘𝐽)‘(𝑓𝑧)))
4847sseq1d 3773 . . . . . . . . . 10 (𝑤 = (𝑓𝑧) → (((cls‘𝐽)‘𝑤) ⊆ (𝑋𝑧) ↔ ((cls‘𝐽)‘(𝑓𝑧)) ⊆ (𝑋𝑧)))
4946, 48anbi12d 749 . . . . . . . . 9 (𝑤 = (𝑓𝑧) → ((𝐴𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑋𝑧)) ↔ (𝐴 ∈ (𝑓𝑧) ∧ ((cls‘𝐽)‘(𝑓𝑧)) ⊆ (𝑋𝑧))))
5049ac6sfi 8369 . . . . . . . 8 ((𝑥 ∈ Fin ∧ ∀𝑧𝑥𝑤𝐽 (𝐴𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑋𝑧))) → ∃𝑓(𝑓:𝑥𝐽 ∧ ∀𝑧𝑥 (𝐴 ∈ (𝑓𝑧) ∧ ((cls‘𝐽)‘(𝑓𝑧)) ⊆ (𝑋𝑧))))
5134, 45, 50syl2anr 496 . . . . . . 7 ((𝑥 ∈ 𝒫 𝑂𝑥 ∈ Fin) → ∃𝑓(𝑓:𝑥𝐽 ∧ ∀𝑧𝑥 (𝐴 ∈ (𝑓𝑧) ∧ ((cls‘𝐽)‘(𝑓𝑧)) ⊆ (𝑋𝑧))))
5233, 51sylbi 207 . . . . . 6 (𝑥 ∈ (𝒫 𝑂 ∩ Fin) → ∃𝑓(𝑓:𝑥𝐽 ∧ ∀𝑧𝑥 (𝐴 ∈ (𝑓𝑧) ∧ ((cls‘𝐽)‘(𝑓𝑧)) ⊆ (𝑋𝑧))))
5352ad2antlr 765 . . . . 5 (((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ (𝑆 𝑥𝑥 ≠ ∅)) → ∃𝑓(𝑓:𝑥𝐽 ∧ ∀𝑧𝑥 (𝐴 ∈ (𝑓𝑧) ∧ ((cls‘𝐽)‘(𝑓𝑧)) ⊆ (𝑋𝑧))))
543ad3antrrr 768 . . . . . . 7 ((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ (𝑆 𝑥𝑥 ≠ ∅)) ∧ (𝑓:𝑥𝐽 ∧ ∀𝑧𝑥 (𝐴 ∈ (𝑓𝑧) ∧ ((cls‘𝐽)‘(𝑓𝑧)) ⊆ (𝑋𝑧)))) → 𝐽 ∈ Top)
55 frn 6214 . . . . . . . 8 (𝑓:𝑥𝐽 → ran 𝑓𝐽)
5655ad2antrl 766 . . . . . . 7 ((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ (𝑆 𝑥𝑥 ≠ ∅)) ∧ (𝑓:𝑥𝐽 ∧ ∀𝑧𝑥 (𝐴 ∈ (𝑓𝑧) ∧ ((cls‘𝐽)‘(𝑓𝑧)) ⊆ (𝑋𝑧)))) → ran 𝑓𝐽)
57 simprr 813 . . . . . . . 8 (((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ (𝑆 𝑥𝑥 ≠ ∅)) → 𝑥 ≠ ∅)
58 simpl 474 . . . . . . . 8 ((𝑓:𝑥𝐽 ∧ ∀𝑧𝑥 (𝐴 ∈ (𝑓𝑧) ∧ ((cls‘𝐽)‘(𝑓𝑧)) ⊆ (𝑋𝑧))) → 𝑓:𝑥𝐽)
59 dm0rn0 5497 . . . . . . . . . . 11 (dom 𝑓 = ∅ ↔ ran 𝑓 = ∅)
60 fdm 6212 . . . . . . . . . . . 12 (𝑓:𝑥𝐽 → dom 𝑓 = 𝑥)
6160eqeq1d 2762 . . . . . . . . . . 11 (𝑓:𝑥𝐽 → (dom 𝑓 = ∅ ↔ 𝑥 = ∅))
6259, 61syl5rbbr 275 . . . . . . . . . 10 (𝑓:𝑥𝐽 → (𝑥 = ∅ ↔ ran 𝑓 = ∅))
6362necon3bid 2976 . . . . . . . . 9 (𝑓:𝑥𝐽 → (𝑥 ≠ ∅ ↔ ran 𝑓 ≠ ∅))
6463biimpac 504 . . . . . . . 8 ((𝑥 ≠ ∅ ∧ 𝑓:𝑥𝐽) → ran 𝑓 ≠ ∅)
6557, 58, 64syl2an 495 . . . . . . 7 ((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ (𝑆 𝑥𝑥 ≠ ∅)) ∧ (𝑓:𝑥𝐽 ∧ ∀𝑧𝑥 (𝐴 ∈ (𝑓𝑧) ∧ ((cls‘𝐽)‘(𝑓𝑧)) ⊆ (𝑋𝑧)))) → ran 𝑓 ≠ ∅)
6633simprbi 483 . . . . . . . . 9 (𝑥 ∈ (𝒫 𝑂 ∩ Fin) → 𝑥 ∈ Fin)
6766ad2antlr 765 . . . . . . . 8 (((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ (𝑆 𝑥𝑥 ≠ ∅)) → 𝑥 ∈ Fin)
68 ffn 6206 . . . . . . . . . 10 (𝑓:𝑥𝐽𝑓 Fn 𝑥)
69 dffn4 6282 . . . . . . . . . 10 (𝑓 Fn 𝑥𝑓:𝑥onto→ran 𝑓)
7068, 69sylib 208 . . . . . . . . 9 (𝑓:𝑥𝐽𝑓:𝑥onto→ran 𝑓)
7170adantr 472 . . . . . . . 8 ((𝑓:𝑥𝐽 ∧ ∀𝑧𝑥 (𝐴 ∈ (𝑓𝑧) ∧ ((cls‘𝐽)‘(𝑓𝑧)) ⊆ (𝑋𝑧))) → 𝑓:𝑥onto→ran 𝑓)
72 fofi 8417 . . . . . . . 8 ((𝑥 ∈ Fin ∧ 𝑓:𝑥onto→ran 𝑓) → ran 𝑓 ∈ Fin)
7367, 71, 72syl2an 495 . . . . . . 7 ((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ (𝑆 𝑥𝑥 ≠ ∅)) ∧ (𝑓:𝑥𝐽 ∧ ∀𝑧𝑥 (𝐴 ∈ (𝑓𝑧) ∧ ((cls‘𝐽)‘(𝑓𝑧)) ⊆ (𝑋𝑧)))) → ran 𝑓 ∈ Fin)
74 fiinopn 20908 . . . . . . . 8 (𝐽 ∈ Top → ((ran 𝑓𝐽 ∧ ran 𝑓 ≠ ∅ ∧ ran 𝑓 ∈ Fin) → ran 𝑓𝐽))
7574imp 444 . . . . . . 7 ((𝐽 ∈ Top ∧ (ran 𝑓𝐽 ∧ ran 𝑓 ≠ ∅ ∧ ran 𝑓 ∈ Fin)) → ran 𝑓𝐽)
7654, 56, 65, 73, 75syl13anc 1479 . . . . . 6 ((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ (𝑆 𝑥𝑥 ≠ ∅)) ∧ (𝑓:𝑥𝐽 ∧ ∀𝑧𝑥 (𝐴 ∈ (𝑓𝑧) ∧ ((cls‘𝐽)‘(𝑓𝑧)) ⊆ (𝑋𝑧)))) → ran 𝑓𝐽)
77 simpl 474 . . . . . . . . . 10 ((𝐴 ∈ (𝑓𝑧) ∧ ((cls‘𝐽)‘(𝑓𝑧)) ⊆ (𝑋𝑧)) → 𝐴 ∈ (𝑓𝑧))
7877ralimi 3090 . . . . . . . . 9 (∀𝑧𝑥 (𝐴 ∈ (𝑓𝑧) ∧ ((cls‘𝐽)‘(𝑓𝑧)) ⊆ (𝑋𝑧)) → ∀𝑧𝑥 𝐴 ∈ (𝑓𝑧))
7978ad2antll 767 . . . . . . . 8 ((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ (𝑆 𝑥𝑥 ≠ ∅)) ∧ (𝑓:𝑥𝐽 ∧ ∀𝑧𝑥 (𝐴 ∈ (𝑓𝑧) ∧ ((cls‘𝐽)‘(𝑓𝑧)) ⊆ (𝑋𝑧)))) → ∀𝑧𝑥 𝐴 ∈ (𝑓𝑧))
808ad3antrrr 768 . . . . . . . . 9 ((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ (𝑆 𝑥𝑥 ≠ ∅)) ∧ (𝑓:𝑥𝐽 ∧ ∀𝑧𝑥 (𝐴 ∈ (𝑓𝑧) ∧ ((cls‘𝐽)‘(𝑓𝑧)) ⊆ (𝑋𝑧)))) → 𝐴 ∈ (𝑋𝑆))
81 eliin 4677 . . . . . . . . 9 (𝐴 ∈ (𝑋𝑆) → (𝐴 𝑧𝑥 (𝑓𝑧) ↔ ∀𝑧𝑥 𝐴 ∈ (𝑓𝑧)))
8280, 81syl 17 . . . . . . . 8 ((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ (𝑆 𝑥𝑥 ≠ ∅)) ∧ (𝑓:𝑥𝐽 ∧ ∀𝑧𝑥 (𝐴 ∈ (𝑓𝑧) ∧ ((cls‘𝐽)‘(𝑓𝑧)) ⊆ (𝑋𝑧)))) → (𝐴 𝑧𝑥 (𝑓𝑧) ↔ ∀𝑧𝑥 𝐴 ∈ (𝑓𝑧)))
8379, 82mpbird 247 . . . . . . 7 ((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ (𝑆 𝑥𝑥 ≠ ∅)) ∧ (𝑓:𝑥𝐽 ∧ ∀𝑧𝑥 (𝐴 ∈ (𝑓𝑧) ∧ ((cls‘𝐽)‘(𝑓𝑧)) ⊆ (𝑋𝑧)))) → 𝐴 𝑧𝑥 (𝑓𝑧))
8468ad2antrl 766 . . . . . . . 8 ((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ (𝑆 𝑥𝑥 ≠ ∅)) ∧ (𝑓:𝑥𝐽 ∧ ∀𝑧𝑥 (𝐴 ∈ (𝑓𝑧) ∧ ((cls‘𝐽)‘(𝑓𝑧)) ⊆ (𝑋𝑧)))) → 𝑓 Fn 𝑥)
85 fnrnfv 6404 . . . . . . . . . 10 (𝑓 Fn 𝑥 → ran 𝑓 = {𝑦 ∣ ∃𝑧𝑥 𝑦 = (𝑓𝑧)})
8685inteqd 4632 . . . . . . . . 9 (𝑓 Fn 𝑥 ran 𝑓 = {𝑦 ∣ ∃𝑧𝑥 𝑦 = (𝑓𝑧)})
87 fvex 6362 . . . . . . . . . 10 (𝑓𝑧) ∈ V
8887dfiin2 4707 . . . . . . . . 9 𝑧𝑥 (𝑓𝑧) = {𝑦 ∣ ∃𝑧𝑥 𝑦 = (𝑓𝑧)}
8986, 88syl6eqr 2812 . . . . . . . 8 (𝑓 Fn 𝑥 ran 𝑓 = 𝑧𝑥 (𝑓𝑧))
9084, 89syl 17 . . . . . . 7 ((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ (𝑆 𝑥𝑥 ≠ ∅)) ∧ (𝑓:𝑥𝐽 ∧ ∀𝑧𝑥 (𝐴 ∈ (𝑓𝑧) ∧ ((cls‘𝐽)‘(𝑓𝑧)) ⊆ (𝑋𝑧)))) → ran 𝑓 = 𝑧𝑥 (𝑓𝑧))
9183, 90eleqtrrd 2842 . . . . . 6 ((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ (𝑆 𝑥𝑥 ≠ ∅)) ∧ (𝑓:𝑥𝐽 ∧ ∀𝑧𝑥 (𝐴 ∈ (𝑓𝑧) ∧ ((cls‘𝐽)‘(𝑓𝑧)) ⊆ (𝑋𝑧)))) → 𝐴 ran 𝑓)
9257adantr 472 . . . . . . . . 9 ((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ (𝑆 𝑥𝑥 ≠ ∅)) ∧ (𝑓:𝑥𝐽 ∧ ∀𝑧𝑥 (𝐴 ∈ (𝑓𝑧) ∧ ((cls‘𝐽)‘(𝑓𝑧)) ⊆ (𝑋𝑧)))) → 𝑥 ≠ ∅)
933ad4antr 771 . . . . . . . . . . . 12 (((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ (𝑆 𝑥𝑥 ≠ ∅)) ∧ 𝑓:𝑥𝐽) ∧ 𝑧𝑥) → 𝐽 ∈ Top)
94 ffvelrn 6520 . . . . . . . . . . . . . . 15 ((𝑓:𝑥𝐽𝑧𝑥) → (𝑓𝑧) ∈ 𝐽)
9594adantll 752 . . . . . . . . . . . . . 14 (((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ (𝑆 𝑥𝑥 ≠ ∅)) ∧ 𝑓:𝑥𝐽) ∧ 𝑧𝑥) → (𝑓𝑧) ∈ 𝐽)
96 elssuni 4619 . . . . . . . . . . . . . 14 ((𝑓𝑧) ∈ 𝐽 → (𝑓𝑧) ⊆ 𝐽)
9795, 96syl 17 . . . . . . . . . . . . 13 (((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ (𝑆 𝑥𝑥 ≠ ∅)) ∧ 𝑓:𝑥𝐽) ∧ 𝑧𝑥) → (𝑓𝑧) ⊆ 𝐽)
9897, 5syl6sseqr 3793 . . . . . . . . . . . 12 (((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ (𝑆 𝑥𝑥 ≠ ∅)) ∧ 𝑓:𝑥𝐽) ∧ 𝑧𝑥) → (𝑓𝑧) ⊆ 𝑋)
995clscld 21053 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ (𝑓𝑧) ⊆ 𝑋) → ((cls‘𝐽)‘(𝑓𝑧)) ∈ (Clsd‘𝐽))
10093, 98, 99syl2anc 696 . . . . . . . . . . 11 (((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ (𝑆 𝑥𝑥 ≠ ∅)) ∧ 𝑓:𝑥𝐽) ∧ 𝑧𝑥) → ((cls‘𝐽)‘(𝑓𝑧)) ∈ (Clsd‘𝐽))
101100ralrimiva 3104 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ (𝑆 𝑥𝑥 ≠ ∅)) ∧ 𝑓:𝑥𝐽) → ∀𝑧𝑥 ((cls‘𝐽)‘(𝑓𝑧)) ∈ (Clsd‘𝐽))
102101adantrr 755 . . . . . . . . 9 ((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ (𝑆 𝑥𝑥 ≠ ∅)) ∧ (𝑓:𝑥𝐽 ∧ ∀𝑧𝑥 (𝐴 ∈ (𝑓𝑧) ∧ ((cls‘𝐽)‘(𝑓𝑧)) ⊆ (𝑋𝑧)))) → ∀𝑧𝑥 ((cls‘𝐽)‘(𝑓𝑧)) ∈ (Clsd‘𝐽))
103 iincld 21045 . . . . . . . . 9 ((𝑥 ≠ ∅ ∧ ∀𝑧𝑥 ((cls‘𝐽)‘(𝑓𝑧)) ∈ (Clsd‘𝐽)) → 𝑧𝑥 ((cls‘𝐽)‘(𝑓𝑧)) ∈ (Clsd‘𝐽))
10492, 102, 103syl2anc 696 . . . . . . . 8 ((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ (𝑆 𝑥𝑥 ≠ ∅)) ∧ (𝑓:𝑥𝐽 ∧ ∀𝑧𝑥 (𝐴 ∈ (𝑓𝑧) ∧ ((cls‘𝐽)‘(𝑓𝑧)) ⊆ (𝑋𝑧)))) → 𝑧𝑥 ((cls‘𝐽)‘(𝑓𝑧)) ∈ (Clsd‘𝐽))
1055sscls 21062 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ (𝑓𝑧) ⊆ 𝑋) → (𝑓𝑧) ⊆ ((cls‘𝐽)‘(𝑓𝑧)))
10693, 98, 105syl2anc 696 . . . . . . . . . . . 12 (((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ (𝑆 𝑥𝑥 ≠ ∅)) ∧ 𝑓:𝑥𝐽) ∧ 𝑧𝑥) → (𝑓𝑧) ⊆ ((cls‘𝐽)‘(𝑓𝑧)))
107106ralrimiva 3104 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ (𝑆 𝑥𝑥 ≠ ∅)) ∧ 𝑓:𝑥𝐽) → ∀𝑧𝑥 (𝑓𝑧) ⊆ ((cls‘𝐽)‘(𝑓𝑧)))
108 ssel 3738 . . . . . . . . . . . . . 14 ((𝑓𝑧) ⊆ ((cls‘𝐽)‘(𝑓𝑧)) → (𝑦 ∈ (𝑓𝑧) → 𝑦 ∈ ((cls‘𝐽)‘(𝑓𝑧))))
109108ral2imi 3085 . . . . . . . . . . . . 13 (∀𝑧𝑥 (𝑓𝑧) ⊆ ((cls‘𝐽)‘(𝑓𝑧)) → (∀𝑧𝑥 𝑦 ∈ (𝑓𝑧) → ∀𝑧𝑥 𝑦 ∈ ((cls‘𝐽)‘(𝑓𝑧))))
110 vex 3343 . . . . . . . . . . . . . 14 𝑦 ∈ V
111 eliin 4677 . . . . . . . . . . . . . 14 (𝑦 ∈ V → (𝑦 𝑧𝑥 (𝑓𝑧) ↔ ∀𝑧𝑥 𝑦 ∈ (𝑓𝑧)))
112110, 111ax-mp 5 . . . . . . . . . . . . 13 (𝑦 𝑧𝑥 (𝑓𝑧) ↔ ∀𝑧𝑥 𝑦 ∈ (𝑓𝑧))
113 eliin 4677 . . . . . . . . . . . . . 14 (𝑦 ∈ V → (𝑦 𝑧𝑥 ((cls‘𝐽)‘(𝑓𝑧)) ↔ ∀𝑧𝑥 𝑦 ∈ ((cls‘𝐽)‘(𝑓𝑧))))
114110, 113ax-mp 5 . . . . . . . . . . . . 13 (𝑦 𝑧𝑥 ((cls‘𝐽)‘(𝑓𝑧)) ↔ ∀𝑧𝑥 𝑦 ∈ ((cls‘𝐽)‘(𝑓𝑧)))
115109, 112, 1143imtr4g 285 . . . . . . . . . . . 12 (∀𝑧𝑥 (𝑓𝑧) ⊆ ((cls‘𝐽)‘(𝑓𝑧)) → (𝑦 𝑧𝑥 (𝑓𝑧) → 𝑦 𝑧𝑥 ((cls‘𝐽)‘(𝑓𝑧))))
116115ssrdv 3750 . . . . . . . . . . 11 (∀𝑧𝑥 (𝑓𝑧) ⊆ ((cls‘𝐽)‘(𝑓𝑧)) → 𝑧𝑥 (𝑓𝑧) ⊆ 𝑧𝑥 ((cls‘𝐽)‘(𝑓𝑧)))
117107, 116syl 17 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ (𝑆 𝑥𝑥 ≠ ∅)) ∧ 𝑓:𝑥𝐽) → 𝑧𝑥 (𝑓𝑧) ⊆ 𝑧𝑥 ((cls‘𝐽)‘(𝑓𝑧)))
118117adantrr 755 . . . . . . . . 9 ((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ (𝑆 𝑥𝑥 ≠ ∅)) ∧ (𝑓:𝑥𝐽 ∧ ∀𝑧𝑥 (𝐴 ∈ (𝑓𝑧) ∧ ((cls‘𝐽)‘(𝑓𝑧)) ⊆ (𝑋𝑧)))) → 𝑧𝑥 (𝑓𝑧) ⊆ 𝑧𝑥 ((cls‘𝐽)‘(𝑓𝑧)))
11990, 118eqsstrd 3780 . . . . . . . 8 ((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ (𝑆 𝑥𝑥 ≠ ∅)) ∧ (𝑓:𝑥𝐽 ∧ ∀𝑧𝑥 (𝐴 ∈ (𝑓𝑧) ∧ ((cls‘𝐽)‘(𝑓𝑧)) ⊆ (𝑋𝑧)))) → ran 𝑓 𝑧𝑥 ((cls‘𝐽)‘(𝑓𝑧)))
1205clsss2 21078 . . . . . . . 8 (( 𝑧𝑥 ((cls‘𝐽)‘(𝑓𝑧)) ∈ (Clsd‘𝐽) ∧ ran 𝑓 𝑧𝑥 ((cls‘𝐽)‘(𝑓𝑧))) → ((cls‘𝐽)‘ ran 𝑓) ⊆ 𝑧𝑥 ((cls‘𝐽)‘(𝑓𝑧)))
121104, 119, 120syl2anc 696 . . . . . . 7 ((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ (𝑆 𝑥𝑥 ≠ ∅)) ∧ (𝑓:𝑥𝐽 ∧ ∀𝑧𝑥 (𝐴 ∈ (𝑓𝑧) ∧ ((cls‘𝐽)‘(𝑓𝑧)) ⊆ (𝑋𝑧)))) → ((cls‘𝐽)‘ ran 𝑓) ⊆ 𝑧𝑥 ((cls‘𝐽)‘(𝑓𝑧)))
122 ssel 3738 . . . . . . . . . . . . 13 (((cls‘𝐽)‘(𝑓𝑧)) ⊆ (𝑋𝑧) → (𝑦 ∈ ((cls‘𝐽)‘(𝑓𝑧)) → 𝑦 ∈ (𝑋𝑧)))
123122adantl 473 . . . . . . . . . . . 12 ((𝐴 ∈ (𝑓𝑧) ∧ ((cls‘𝐽)‘(𝑓𝑧)) ⊆ (𝑋𝑧)) → (𝑦 ∈ ((cls‘𝐽)‘(𝑓𝑧)) → 𝑦 ∈ (𝑋𝑧)))
124123ral2imi 3085 . . . . . . . . . . 11 (∀𝑧𝑥 (𝐴 ∈ (𝑓𝑧) ∧ ((cls‘𝐽)‘(𝑓𝑧)) ⊆ (𝑋𝑧)) → (∀𝑧𝑥 𝑦 ∈ ((cls‘𝐽)‘(𝑓𝑧)) → ∀𝑧𝑥 𝑦 ∈ (𝑋𝑧)))
125 eliin 4677 . . . . . . . . . . . 12 (𝑦 ∈ V → (𝑦 𝑧𝑥 (𝑋𝑧) ↔ ∀𝑧𝑥 𝑦 ∈ (𝑋𝑧)))
126110, 125ax-mp 5 . . . . . . . . . . 11 (𝑦 𝑧𝑥 (𝑋𝑧) ↔ ∀𝑧𝑥 𝑦 ∈ (𝑋𝑧))
127124, 114, 1263imtr4g 285 . . . . . . . . . 10 (∀𝑧𝑥 (𝐴 ∈ (𝑓𝑧) ∧ ((cls‘𝐽)‘(𝑓𝑧)) ⊆ (𝑋𝑧)) → (𝑦 𝑧𝑥 ((cls‘𝐽)‘(𝑓𝑧)) → 𝑦 𝑧𝑥 (𝑋𝑧)))
128127ssrdv 3750 . . . . . . . . 9 (∀𝑧𝑥 (𝐴 ∈ (𝑓𝑧) ∧ ((cls‘𝐽)‘(𝑓𝑧)) ⊆ (𝑋𝑧)) → 𝑧𝑥 ((cls‘𝐽)‘(𝑓𝑧)) ⊆ 𝑧𝑥 (𝑋𝑧))
129128ad2antll 767 . . . . . . . 8 ((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ (𝑆 𝑥𝑥 ≠ ∅)) ∧ (𝑓:𝑥𝐽 ∧ ∀𝑧𝑥 (𝐴 ∈ (𝑓𝑧) ∧ ((cls‘𝐽)‘(𝑓𝑧)) ⊆ (𝑋𝑧)))) → 𝑧𝑥 ((cls‘𝐽)‘(𝑓𝑧)) ⊆ 𝑧𝑥 (𝑋𝑧))
130 iindif2 4741 . . . . . . . . . 10 (𝑥 ≠ ∅ → 𝑧𝑥 (𝑋𝑧) = (𝑋 𝑧𝑥 𝑧))
13192, 130syl 17 . . . . . . . . 9 ((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ (𝑆 𝑥𝑥 ≠ ∅)) ∧ (𝑓:𝑥𝐽 ∧ ∀𝑧𝑥 (𝐴 ∈ (𝑓𝑧) ∧ ((cls‘𝐽)‘(𝑓𝑧)) ⊆ (𝑋𝑧)))) → 𝑧𝑥 (𝑋𝑧) = (𝑋 𝑧𝑥 𝑧))
132 simplrl 819 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ (𝑆 𝑥𝑥 ≠ ∅)) ∧ (𝑓:𝑥𝐽 ∧ ∀𝑧𝑥 (𝐴 ∈ (𝑓𝑧) ∧ ((cls‘𝐽)‘(𝑓𝑧)) ⊆ (𝑋𝑧)))) → 𝑆 𝑥)
133 uniiun 4725 . . . . . . . . . . . 12 𝑥 = 𝑧𝑥 𝑧
134133sseq2i 3771 . . . . . . . . . . 11 (𝑆 𝑥𝑆 𝑧𝑥 𝑧)
135 sscon 3887 . . . . . . . . . . 11 (𝑆 𝑧𝑥 𝑧 → (𝑋 𝑧𝑥 𝑧) ⊆ (𝑋𝑆))
136134, 135sylbi 207 . . . . . . . . . 10 (𝑆 𝑥 → (𝑋 𝑧𝑥 𝑧) ⊆ (𝑋𝑆))
137132, 136syl 17 . . . . . . . . 9 ((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ (𝑆 𝑥𝑥 ≠ ∅)) ∧ (𝑓:𝑥𝐽 ∧ ∀𝑧𝑥 (𝐴 ∈ (𝑓𝑧) ∧ ((cls‘𝐽)‘(𝑓𝑧)) ⊆ (𝑋𝑧)))) → (𝑋 𝑧𝑥 𝑧) ⊆ (𝑋𝑆))
138131, 137eqsstrd 3780 . . . . . . . 8 ((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ (𝑆 𝑥𝑥 ≠ ∅)) ∧ (𝑓:𝑥𝐽 ∧ ∀𝑧𝑥 (𝐴 ∈ (𝑓𝑧) ∧ ((cls‘𝐽)‘(𝑓𝑧)) ⊆ (𝑋𝑧)))) → 𝑧𝑥 (𝑋𝑧) ⊆ (𝑋𝑆))
139129, 138sstrd 3754 . . . . . . 7 ((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ (𝑆 𝑥𝑥 ≠ ∅)) ∧ (𝑓:𝑥𝐽 ∧ ∀𝑧𝑥 (𝐴 ∈ (𝑓𝑧) ∧ ((cls‘𝐽)‘(𝑓𝑧)) ⊆ (𝑋𝑧)))) → 𝑧𝑥 ((cls‘𝐽)‘(𝑓𝑧)) ⊆ (𝑋𝑆))
140121, 139sstrd 3754 . . . . . 6 ((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ (𝑆 𝑥𝑥 ≠ ∅)) ∧ (𝑓:𝑥𝐽 ∧ ∀𝑧𝑥 (𝐴 ∈ (𝑓𝑧) ∧ ((cls‘𝐽)‘(𝑓𝑧)) ⊆ (𝑋𝑧)))) → ((cls‘𝐽)‘ ran 𝑓) ⊆ (𝑋𝑆))
141 eleq2 2828 . . . . . . . 8 (𝑧 = ran 𝑓 → (𝐴𝑧𝐴 ran 𝑓))
142 fveq2 6352 . . . . . . . . 9 (𝑧 = ran 𝑓 → ((cls‘𝐽)‘𝑧) = ((cls‘𝐽)‘ ran 𝑓))
143142sseq1d 3773 . . . . . . . 8 (𝑧 = ran 𝑓 → (((cls‘𝐽)‘𝑧) ⊆ (𝑋𝑆) ↔ ((cls‘𝐽)‘ ran 𝑓) ⊆ (𝑋𝑆)))
144141, 143anbi12d 749 . . . . . . 7 (𝑧 = ran 𝑓 → ((𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ (𝑋𝑆)) ↔ (𝐴 ran 𝑓 ∧ ((cls‘𝐽)‘ ran 𝑓) ⊆ (𝑋𝑆))))
145144rspcev 3449 . . . . . 6 (( ran 𝑓𝐽 ∧ (𝐴 ran 𝑓 ∧ ((cls‘𝐽)‘ ran 𝑓) ⊆ (𝑋𝑆))) → ∃𝑧𝐽 (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ (𝑋𝑆)))
14676, 91, 140, 145syl12anc 1475 . . . . 5 ((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ (𝑆 𝑥𝑥 ≠ ∅)) ∧ (𝑓:𝑥𝐽 ∧ ∀𝑧𝑥 (𝐴 ∈ (𝑓𝑧) ∧ ((cls‘𝐽)‘(𝑓𝑧)) ⊆ (𝑋𝑧)))) → ∃𝑧𝐽 (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ (𝑋𝑆)))
14753, 146exlimddv 2012 . . . 4 (((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ (𝑆 𝑥𝑥 ≠ ∅)) → ∃𝑧𝐽 (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ (𝑋𝑆)))
148147anassrs 683 . . 3 ((((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ 𝑆 𝑥) ∧ 𝑥 ≠ ∅) → ∃𝑧𝐽 (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ (𝑋𝑆)))
14932, 148pm2.61dane 3019 . 2 (((𝜑𝑥 ∈ (𝒫 𝑂 ∩ Fin)) ∧ 𝑆 𝑥) → ∃𝑧𝐽 (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ (𝑋𝑆)))
1501adantr 472 . . . . . . . 8 ((𝜑𝑥𝑆) → 𝐽 ∈ Haus)
151 hauscmplem.4 . . . . . . . . 9 (𝜑𝑆𝑋)
152151sselda 3744 . . . . . . . 8 ((𝜑𝑥𝑆) → 𝑥𝑋)
1539adantr 472 . . . . . . . 8 ((𝜑𝑥𝑆) → 𝐴𝑋)
154 id 22 . . . . . . . . 9 (𝑥𝑆𝑥𝑆)
1558eldifbd 3728 . . . . . . . . 9 (𝜑 → ¬ 𝐴𝑆)
156 nelne2 3029 . . . . . . . . 9 ((𝑥𝑆 ∧ ¬ 𝐴𝑆) → 𝑥𝐴)
157154, 155, 156syl2anr 496 . . . . . . . 8 ((𝜑𝑥𝑆) → 𝑥𝐴)
1585hausnei 21334 . . . . . . . 8 ((𝐽 ∈ Haus ∧ (𝑥𝑋𝐴𝑋𝑥𝐴)) → ∃𝑦𝐽𝑤𝐽 (𝑥𝑦𝐴𝑤 ∧ (𝑦𝑤) = ∅))
159150, 152, 153, 157, 158syl13anc 1479 . . . . . . 7 ((𝜑𝑥𝑆) → ∃𝑦𝐽𝑤𝐽 (𝑥𝑦𝐴𝑤 ∧ (𝑦𝑤) = ∅))
160 3anass 1081 . . . . . . . . . . 11 ((𝑥𝑦𝐴𝑤 ∧ (𝑦𝑤) = ∅) ↔ (𝑥𝑦 ∧ (𝐴𝑤 ∧ (𝑦𝑤) = ∅)))
161 elssuni 4619 . . . . . . . . . . . . . . . . 17 (𝑤𝐽𝑤 𝐽)
162161, 5syl6sseqr 3793 . . . . . . . . . . . . . . . 16 (𝑤𝐽𝑤𝑋)
163162adantl 473 . . . . . . . . . . . . . . 15 ((((𝜑𝑥𝑆) ∧ 𝑦𝐽) ∧ 𝑤𝐽) → 𝑤𝑋)
164 incom 3948 . . . . . . . . . . . . . . . . 17 (𝑦𝑤) = (𝑤𝑦)
165164eqeq1i 2765 . . . . . . . . . . . . . . . 16 ((𝑦𝑤) = ∅ ↔ (𝑤𝑦) = ∅)
166 reldisj 4163 . . . . . . . . . . . . . . . 16 (𝑤𝑋 → ((𝑤𝑦) = ∅ ↔ 𝑤 ⊆ (𝑋𝑦)))
167165, 166syl5bb 272 . . . . . . . . . . . . . . 15 (𝑤𝑋 → ((𝑦𝑤) = ∅ ↔ 𝑤 ⊆ (𝑋𝑦)))
168163, 167syl 17 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝑆) ∧ 𝑦𝐽) ∧ 𝑤𝐽) → ((𝑦𝑤) = ∅ ↔ 𝑤 ⊆ (𝑋𝑦)))
169150, 2syl 17 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝑆) → 𝐽 ∈ Top)
1705opncld 21039 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ Top ∧ 𝑦𝐽) → (𝑋𝑦) ∈ (Clsd‘𝐽))
171169, 170sylan 489 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝑆) ∧ 𝑦𝐽) → (𝑋𝑦) ∈ (Clsd‘𝐽))
172171adantr 472 . . . . . . . . . . . . . . 15 ((((𝜑𝑥𝑆) ∧ 𝑦𝐽) ∧ 𝑤𝐽) → (𝑋𝑦) ∈ (Clsd‘𝐽))
1735clsss2 21078 . . . . . . . . . . . . . . . 16 (((𝑋𝑦) ∈ (Clsd‘𝐽) ∧ 𝑤 ⊆ (𝑋𝑦)) → ((cls‘𝐽)‘𝑤) ⊆ (𝑋𝑦))
174173ex 449 . . . . . . . . . . . . . . 15 ((𝑋𝑦) ∈ (Clsd‘𝐽) → (𝑤 ⊆ (𝑋𝑦) → ((cls‘𝐽)‘𝑤) ⊆ (𝑋𝑦)))
175172, 174syl 17 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝑆) ∧ 𝑦𝐽) ∧ 𝑤𝐽) → (𝑤 ⊆ (𝑋𝑦) → ((cls‘𝐽)‘𝑤) ⊆ (𝑋𝑦)))
176168, 175sylbid 230 . . . . . . . . . . . . 13 ((((𝜑𝑥𝑆) ∧ 𝑦𝐽) ∧ 𝑤𝐽) → ((𝑦𝑤) = ∅ → ((cls‘𝐽)‘𝑤) ⊆ (𝑋𝑦)))
177176anim2d 590 . . . . . . . . . . . 12 ((((𝜑𝑥𝑆) ∧ 𝑦𝐽) ∧ 𝑤𝐽) → ((𝐴𝑤 ∧ (𝑦𝑤) = ∅) → (𝐴𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑋𝑦))))
178177anim2d 590 . . . . . . . . . . 11 ((((𝜑𝑥𝑆) ∧ 𝑦𝐽) ∧ 𝑤𝐽) → ((𝑥𝑦 ∧ (𝐴𝑤 ∧ (𝑦𝑤) = ∅)) → (𝑥𝑦 ∧ (𝐴𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑋𝑦)))))
179160, 178syl5bi 232 . . . . . . . . . 10 ((((𝜑𝑥𝑆) ∧ 𝑦𝐽) ∧ 𝑤𝐽) → ((𝑥𝑦𝐴𝑤 ∧ (𝑦𝑤) = ∅) → (𝑥𝑦 ∧ (𝐴𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑋𝑦)))))
180179reximdva 3155 . . . . . . . . 9 (((𝜑𝑥𝑆) ∧ 𝑦𝐽) → (∃𝑤𝐽 (𝑥𝑦𝐴𝑤 ∧ (𝑦𝑤) = ∅) → ∃𝑤𝐽 (𝑥𝑦 ∧ (𝐴𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑋𝑦)))))
181 r19.42v 3230 . . . . . . . . 9 (∃𝑤𝐽 (𝑥𝑦 ∧ (𝐴𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑋𝑦))) ↔ (𝑥𝑦 ∧ ∃𝑤𝐽 (𝐴𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑋𝑦))))
182180, 181syl6ib 241 . . . . . . . 8 (((𝜑𝑥𝑆) ∧ 𝑦𝐽) → (∃𝑤𝐽 (𝑥𝑦𝐴𝑤 ∧ (𝑦𝑤) = ∅) → (𝑥𝑦 ∧ ∃𝑤𝐽 (𝐴𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑋𝑦)))))
183182reximdva 3155 . . . . . . 7 ((𝜑𝑥𝑆) → (∃𝑦𝐽𝑤𝐽 (𝑥𝑦𝐴𝑤 ∧ (𝑦𝑤) = ∅) → ∃𝑦𝐽 (𝑥𝑦 ∧ ∃𝑤𝐽 (𝐴𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑋𝑦)))))
184159, 183mpd 15 . . . . . 6 ((𝜑𝑥𝑆) → ∃𝑦𝐽 (𝑥𝑦 ∧ ∃𝑤𝐽 (𝐴𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑋𝑦))))
18541unieqi 4597 . . . . . . . 8 𝑂 = {𝑦𝐽 ∣ ∃𝑤𝐽 (𝐴𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑋𝑦))}
186185eleq2i 2831 . . . . . . 7 (𝑥 𝑂𝑥 {𝑦𝐽 ∣ ∃𝑤𝐽 (𝐴𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑋𝑦))})
187 elunirab 4600 . . . . . . 7 (𝑥 {𝑦𝐽 ∣ ∃𝑤𝐽 (𝐴𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑋𝑦))} ↔ ∃𝑦𝐽 (𝑥𝑦 ∧ ∃𝑤𝐽 (𝐴𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑋𝑦))))
188186, 187bitri 264 . . . . . 6 (𝑥 𝑂 ↔ ∃𝑦𝐽 (𝑥𝑦 ∧ ∃𝑤𝐽 (𝐴𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑋𝑦))))
189184, 188sylibr 224 . . . . 5 ((𝜑𝑥𝑆) → 𝑥 𝑂)
190189ex 449 . . . 4 (𝜑 → (𝑥𝑆𝑥 𝑂))
191190ssrdv 3750 . . 3 (𝜑𝑆 𝑂)
192 unieq 4596 . . . . . 6 (𝑧 = 𝑂 𝑧 = 𝑂)
193192sseq2d 3774 . . . . 5 (𝑧 = 𝑂 → (𝑆 𝑧𝑆 𝑂))
194 pweq 4305 . . . . . . 7 (𝑧 = 𝑂 → 𝒫 𝑧 = 𝒫 𝑂)
195194ineq1d 3956 . . . . . 6 (𝑧 = 𝑂 → (𝒫 𝑧 ∩ Fin) = (𝒫 𝑂 ∩ Fin))
196195rexeqdv 3284 . . . . 5 (𝑧 = 𝑂 → (∃𝑥 ∈ (𝒫 𝑧 ∩ Fin)𝑆 𝑥 ↔ ∃𝑥 ∈ (𝒫 𝑂 ∩ Fin)𝑆 𝑥))
197193, 196imbi12d 333 . . . 4 (𝑧 = 𝑂 → ((𝑆 𝑧 → ∃𝑥 ∈ (𝒫 𝑧 ∩ Fin)𝑆 𝑥) ↔ (𝑆 𝑂 → ∃𝑥 ∈ (𝒫 𝑂 ∩ Fin)𝑆 𝑥)))
198 hauscmplem.5 . . . . 5 (𝜑 → (𝐽t 𝑆) ∈ Comp)
1995cmpsub 21405 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝐽t 𝑆) ∈ Comp ↔ ∀𝑧 ∈ 𝒫 𝐽(𝑆 𝑧 → ∃𝑥 ∈ (𝒫 𝑧 ∩ Fin)𝑆 𝑥)))
200199biimp3a 1581 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) → ∀𝑧 ∈ 𝒫 𝐽(𝑆 𝑧 → ∃𝑥 ∈ (𝒫 𝑧 ∩ Fin)𝑆 𝑥))
2013, 151, 198, 200syl3anc 1477 . . . 4 (𝜑 → ∀𝑧 ∈ 𝒫 𝐽(𝑆 𝑧 → ∃𝑥 ∈ (𝒫 𝑧 ∩ Fin)𝑆 𝑥))
202 ssrab2 3828 . . . . . 6 {𝑦𝐽 ∣ ∃𝑤𝐽 (𝐴𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑋𝑦))} ⊆ 𝐽
20341, 202eqsstri 3776 . . . . 5 𝑂𝐽
204 elpw2g 4976 . . . . . 6 (𝐽 ∈ Haus → (𝑂 ∈ 𝒫 𝐽𝑂𝐽))
2051, 204syl 17 . . . . 5 (𝜑 → (𝑂 ∈ 𝒫 𝐽𝑂𝐽))
206203, 205mpbiri 248 . . . 4 (𝜑𝑂 ∈ 𝒫 𝐽)
207197, 201, 206rspcdva 3455 . . 3 (𝜑 → (𝑆 𝑂 → ∃𝑥 ∈ (𝒫 𝑂 ∩ Fin)𝑆 𝑥))
208191, 207mpd 15 . 2 (𝜑 → ∃𝑥 ∈ (𝒫 𝑂 ∩ Fin)𝑆 𝑥)
209149, 208r19.29a 3216 1 (𝜑 → ∃𝑧𝐽 (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ (𝑋𝑆)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wex 1853  wcel 2139  {cab 2746  wne 2932  wral 3050  wrex 3051  {crab 3054  Vcvv 3340  cdif 3712  cin 3714  wss 3715  c0 4058  𝒫 cpw 4302   cuni 4588   cint 4627   ciun 4672   ciin 4673  dom cdm 5266  ran crn 5267   Fn wfn 6044  wf 6045  ontowfo 6047  cfv 6049  (class class class)co 6813  Fincfn 8121  t crest 16283  Topctop 20900  Clsdccld 21022  clsccl 21024  Hauscha 21314  Compccmp 21391
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-en 8122  df-dom 8123  df-fin 8125  df-fi 8482  df-rest 16285  df-topgen 16306  df-top 20901  df-topon 20918  df-bases 20952  df-cld 21025  df-cls 21027  df-haus 21321  df-cmp 21392
This theorem is referenced by:  hauscmp  21412  hausllycmp  21499
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