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Theorem flimfnfcls 23532
Description: A filter converges to a point iff every finer filter clusters there. Along with fclsfnflim 23531, this theorem illustrates the duality between convergence and clustering. (Contributed by Jeff Hankins, 12-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
Hypothesis
Ref Expression
flimfnfcls.x 𝑋 = βˆͺ 𝐽
Assertion
Ref Expression
flimfnfcls (𝐹 ∈ (Filβ€˜π‘‹) β†’ (𝐴 ∈ (𝐽 fLim 𝐹) ↔ βˆ€π‘” ∈ (Filβ€˜π‘‹)(𝐹 βŠ† 𝑔 β†’ 𝐴 ∈ (𝐽 fClus 𝑔))))
Distinct variable groups:   𝐴,𝑔   𝑔,𝐹   𝑔,𝐽   𝑔,𝑋
Proof of Theorem flimfnfcls
Dummy variables π‘œ π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 flimfcls 23530 . . . . 5 (𝐽 fLim 𝑔) βŠ† (𝐽 fClus 𝑔)
2 flimtop 23469 . . . . . . . . 9 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ 𝐽 ∈ Top)
3 flimfnfcls.x . . . . . . . . . 10 𝑋 = βˆͺ 𝐽
43toptopon 22419 . . . . . . . . 9 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜π‘‹))
52, 4sylib 217 . . . . . . . 8 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
65ad2antrr 725 . . . . . . 7 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑔 ∈ (Filβ€˜π‘‹)) ∧ 𝐹 βŠ† 𝑔) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
7 simplr 768 . . . . . . 7 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑔 ∈ (Filβ€˜π‘‹)) ∧ 𝐹 βŠ† 𝑔) β†’ 𝑔 ∈ (Filβ€˜π‘‹))
8 simpr 486 . . . . . . 7 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑔 ∈ (Filβ€˜π‘‹)) ∧ 𝐹 βŠ† 𝑔) β†’ 𝐹 βŠ† 𝑔)
9 flimss2 23476 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐹 βŠ† 𝑔) β†’ (𝐽 fLim 𝐹) βŠ† (𝐽 fLim 𝑔))
106, 7, 8, 9syl3anc 1372 . . . . . 6 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑔 ∈ (Filβ€˜π‘‹)) ∧ 𝐹 βŠ† 𝑔) β†’ (𝐽 fLim 𝐹) βŠ† (𝐽 fLim 𝑔))
11 simpll 766 . . . . . 6 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑔 ∈ (Filβ€˜π‘‹)) ∧ 𝐹 βŠ† 𝑔) β†’ 𝐴 ∈ (𝐽 fLim 𝐹))
1210, 11sseldd 3984 . . . . 5 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑔 ∈ (Filβ€˜π‘‹)) ∧ 𝐹 βŠ† 𝑔) β†’ 𝐴 ∈ (𝐽 fLim 𝑔))
131, 12sselid 3981 . . . 4 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑔 ∈ (Filβ€˜π‘‹)) ∧ 𝐹 βŠ† 𝑔) β†’ 𝐴 ∈ (𝐽 fClus 𝑔))
1413ex 414 . . 3 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑔 ∈ (Filβ€˜π‘‹)) β†’ (𝐹 βŠ† 𝑔 β†’ 𝐴 ∈ (𝐽 fClus 𝑔)))
1514ralrimiva 3147 . 2 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ βˆ€π‘” ∈ (Filβ€˜π‘‹)(𝐹 βŠ† 𝑔 β†’ 𝐴 ∈ (𝐽 fClus 𝑔)))
16 sseq2 4009 . . . . . 6 (𝑔 = 𝐹 β†’ (𝐹 βŠ† 𝑔 ↔ 𝐹 βŠ† 𝐹))
17 oveq2 7417 . . . . . . 7 (𝑔 = 𝐹 β†’ (𝐽 fClus 𝑔) = (𝐽 fClus 𝐹))
1817eleq2d 2820 . . . . . 6 (𝑔 = 𝐹 β†’ (𝐴 ∈ (𝐽 fClus 𝑔) ↔ 𝐴 ∈ (𝐽 fClus 𝐹)))
1916, 18imbi12d 345 . . . . 5 (𝑔 = 𝐹 β†’ ((𝐹 βŠ† 𝑔 β†’ 𝐴 ∈ (𝐽 fClus 𝑔)) ↔ (𝐹 βŠ† 𝐹 β†’ 𝐴 ∈ (𝐽 fClus 𝐹))))
2019rspcv 3609 . . . 4 (𝐹 ∈ (Filβ€˜π‘‹) β†’ (βˆ€π‘” ∈ (Filβ€˜π‘‹)(𝐹 βŠ† 𝑔 β†’ 𝐴 ∈ (𝐽 fClus 𝑔)) β†’ (𝐹 βŠ† 𝐹 β†’ 𝐴 ∈ (𝐽 fClus 𝐹))))
21 ssid 4005 . . . . . 6 𝐹 βŠ† 𝐹
22 id 22 . . . . . 6 ((𝐹 βŠ† 𝐹 β†’ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ (𝐹 βŠ† 𝐹 β†’ 𝐴 ∈ (𝐽 fClus 𝐹)))
2321, 22mpi 20 . . . . 5 ((𝐹 βŠ† 𝐹 β†’ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ 𝐴 ∈ (𝐽 fClus 𝐹))
24 fclstop 23515 . . . . . 6 (𝐴 ∈ (𝐽 fClus 𝐹) β†’ 𝐽 ∈ Top)
253fclselbas 23520 . . . . . 6 (𝐴 ∈ (𝐽 fClus 𝐹) β†’ 𝐴 ∈ 𝑋)
2624, 25jca 513 . . . . 5 (𝐴 ∈ (𝐽 fClus 𝐹) β†’ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋))
2723, 26syl 17 . . . 4 ((𝐹 βŠ† 𝐹 β†’ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋))
2820, 27syl6 35 . . 3 (𝐹 ∈ (Filβ€˜π‘‹) β†’ (βˆ€π‘” ∈ (Filβ€˜π‘‹)(𝐹 βŠ† 𝑔 β†’ 𝐴 ∈ (𝐽 fClus 𝑔)) β†’ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)))
29 disjdif 4472 . . . . . . . . . . . . . 14 (π‘œ ∩ (𝑋 βˆ– π‘œ)) = βˆ…
30 simpll 766 . . . . . . . . . . . . . . . . . . . . 21 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) β†’ 𝐹 ∈ (Filβ€˜π‘‹))
31 simplrl 776 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) β†’ 𝐽 ∈ Top)
323topopn 22408 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐽 ∈ Top β†’ 𝑋 ∈ 𝐽)
3331, 32syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) β†’ 𝑋 ∈ 𝐽)
34 pwexg 5377 . . . . . . . . . . . . . . . . . . . . . 22 (𝑋 ∈ 𝐽 β†’ 𝒫 𝑋 ∈ V)
35 rabexg 5332 . . . . . . . . . . . . . . . . . . . . . 22 (𝒫 𝑋 ∈ V β†’ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯} ∈ V)
3633, 34, 353syl 18 . . . . . . . . . . . . . . . . . . . . 21 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) β†’ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯} ∈ V)
37 unexg 7736 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯} ∈ V) β†’ (𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯}) ∈ V)
3830, 36, 37syl2anc 585 . . . . . . . . . . . . . . . . . . . 20 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) β†’ (𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯}) ∈ V)
39 ssfii 9414 . . . . . . . . . . . . . . . . . . . 20 ((𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯}) ∈ V β†’ (𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯}) βŠ† (fiβ€˜(𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯})))
4038, 39syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) β†’ (𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯}) βŠ† (fiβ€˜(𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯})))
41 filsspw 23355 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹 ∈ (Filβ€˜π‘‹) β†’ 𝐹 βŠ† 𝒫 𝑋)
42 ssrab2 4078 . . . . . . . . . . . . . . . . . . . . . . . 24 {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯} βŠ† 𝒫 𝑋
4342a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹 ∈ (Filβ€˜π‘‹) β†’ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯} βŠ† 𝒫 𝑋)
4441, 43unssd 4187 . . . . . . . . . . . . . . . . . . . . . 22 (𝐹 ∈ (Filβ€˜π‘‹) β†’ (𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯}) βŠ† 𝒫 𝑋)
4544ad2antrr 725 . . . . . . . . . . . . . . . . . . . . 21 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) β†’ (𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯}) βŠ† 𝒫 𝑋)
46 ssun2 4174 . . . . . . . . . . . . . . . . . . . . . . 23 {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯} βŠ† (𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯})
47 sseq2 4009 . . . . . . . . . . . . . . . . . . . . . . . 24 (π‘₯ = (𝑋 βˆ– π‘œ) β†’ ((𝑋 βˆ– π‘œ) βŠ† π‘₯ ↔ (𝑋 βˆ– π‘œ) βŠ† (𝑋 βˆ– π‘œ)))
48 difss 4132 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑋 βˆ– π‘œ) βŠ† 𝑋
49 elpw2g 5345 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑋 ∈ 𝐽 β†’ ((𝑋 βˆ– π‘œ) ∈ 𝒫 𝑋 ↔ (𝑋 βˆ– π‘œ) βŠ† 𝑋))
5033, 49syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) β†’ ((𝑋 βˆ– π‘œ) ∈ 𝒫 𝑋 ↔ (𝑋 βˆ– π‘œ) βŠ† 𝑋))
5148, 50mpbiri 258 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) β†’ (𝑋 βˆ– π‘œ) ∈ 𝒫 𝑋)
52 ssid 4005 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑋 βˆ– π‘œ) βŠ† (𝑋 βˆ– π‘œ)
5352a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) β†’ (𝑋 βˆ– π‘œ) βŠ† (𝑋 βˆ– π‘œ))
5447, 51, 53elrabd 3686 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) β†’ (𝑋 βˆ– π‘œ) ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯})
5546, 54sselid 3981 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) β†’ (𝑋 βˆ– π‘œ) ∈ (𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯}))
5655ne0d 4336 . . . . . . . . . . . . . . . . . . . . 21 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) β†’ (𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯}) β‰  βˆ…)
57 sseq2 4009 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (π‘₯ = 𝑧 β†’ ((𝑋 βˆ– π‘œ) βŠ† π‘₯ ↔ (𝑋 βˆ– π‘œ) βŠ† 𝑧))
5857elrab 3684 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯} ↔ (𝑧 ∈ 𝒫 𝑋 ∧ (𝑋 βˆ– π‘œ) βŠ† 𝑧))
5958simprbi 498 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧 ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯} β†’ (𝑋 βˆ– π‘œ) βŠ† 𝑧)
6059ad2antll 728 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯})) β†’ (𝑋 βˆ– π‘œ) βŠ† 𝑧)
61 sslin 4235 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑋 βˆ– π‘œ) βŠ† 𝑧 β†’ (𝑦 ∩ (𝑋 βˆ– π‘œ)) βŠ† (𝑦 ∩ 𝑧))
6260, 61syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯})) β†’ (𝑦 ∩ (𝑋 βˆ– π‘œ)) βŠ† (𝑦 ∩ 𝑧))
63 simprrr 781 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) β†’ Β¬ π‘œ ∈ 𝐹)
6463adantr 482 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯})) β†’ Β¬ π‘œ ∈ 𝐹)
65 inssdif0 4370 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑦 ∩ 𝑋) βŠ† π‘œ ↔ (𝑦 ∩ (𝑋 βˆ– π‘œ)) = βˆ…)
66 simplll 774 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯})) β†’ 𝐹 ∈ (Filβ€˜π‘‹))
67 simprl 770 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯})) β†’ 𝑦 ∈ 𝐹)
68 filelss 23356 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝑦 ∈ 𝐹) β†’ 𝑦 βŠ† 𝑋)
6966, 67, 68syl2anc 585 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯})) β†’ 𝑦 βŠ† 𝑋)
70 df-ss 3966 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 βŠ† 𝑋 ↔ (𝑦 ∩ 𝑋) = 𝑦)
7169, 70sylib 217 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯})) β†’ (𝑦 ∩ 𝑋) = 𝑦)
7271sseq1d 4014 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯})) β†’ ((𝑦 ∩ 𝑋) βŠ† π‘œ ↔ 𝑦 βŠ† π‘œ))
7330ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯})) ∧ 𝑦 βŠ† π‘œ) β†’ 𝐹 ∈ (Filβ€˜π‘‹))
74 simplrl 776 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯})) ∧ 𝑦 βŠ† π‘œ) β†’ 𝑦 ∈ 𝐹)
75 elssuni 4942 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (π‘œ ∈ 𝐽 β†’ π‘œ βŠ† βˆͺ 𝐽)
7675, 3sseqtrrdi 4034 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (π‘œ ∈ 𝐽 β†’ π‘œ βŠ† 𝑋)
7776ad2antrl 727 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) β†’ π‘œ βŠ† 𝑋)
7877ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯})) ∧ 𝑦 βŠ† π‘œ) β†’ π‘œ βŠ† 𝑋)
79 simpr 486 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯})) ∧ 𝑦 βŠ† π‘œ) β†’ 𝑦 βŠ† π‘œ)
80 filss 23357 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝑦 ∈ 𝐹 ∧ π‘œ βŠ† 𝑋 ∧ 𝑦 βŠ† π‘œ)) β†’ π‘œ ∈ 𝐹)
8173, 74, 78, 79, 80syl13anc 1373 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯})) ∧ 𝑦 βŠ† π‘œ) β†’ π‘œ ∈ 𝐹)
8281ex 414 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯})) β†’ (𝑦 βŠ† π‘œ β†’ π‘œ ∈ 𝐹))
8372, 82sylbid 239 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯})) β†’ ((𝑦 ∩ 𝑋) βŠ† π‘œ β†’ π‘œ ∈ 𝐹))
8465, 83biimtrrid 242 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯})) β†’ ((𝑦 ∩ (𝑋 βˆ– π‘œ)) = βˆ… β†’ π‘œ ∈ 𝐹))
8584necon3bd 2955 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯})) β†’ (Β¬ π‘œ ∈ 𝐹 β†’ (𝑦 ∩ (𝑋 βˆ– π‘œ)) β‰  βˆ…))
8664, 85mpd 15 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯})) β†’ (𝑦 ∩ (𝑋 βˆ– π‘œ)) β‰  βˆ…)
87 ssn0 4401 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑦 ∩ (𝑋 βˆ– π‘œ)) βŠ† (𝑦 ∩ 𝑧) ∧ (𝑦 ∩ (𝑋 βˆ– π‘œ)) β‰  βˆ…) β†’ (𝑦 ∩ 𝑧) β‰  βˆ…)
8862, 86, 87syl2anc 585 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯})) β†’ (𝑦 ∩ 𝑧) β‰  βˆ…)
8988ralrimivva 3201 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) β†’ βˆ€π‘¦ ∈ 𝐹 βˆ€π‘§ ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯} (𝑦 ∩ 𝑧) β‰  βˆ…)
90 filfbas 23352 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐹 ∈ (Filβ€˜π‘‹) β†’ 𝐹 ∈ (fBasβ€˜π‘‹))
9130, 90syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) β†’ 𝐹 ∈ (fBasβ€˜π‘‹))
9248a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) β†’ (𝑋 βˆ– π‘œ) βŠ† 𝑋)
93 filtop 23359 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝐹 ∈ (Filβ€˜π‘‹) β†’ 𝑋 ∈ 𝐹)
9430, 93syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) β†’ 𝑋 ∈ 𝐹)
95 eleq1 2822 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (π‘œ = 𝑋 β†’ (π‘œ ∈ 𝐹 ↔ 𝑋 ∈ 𝐹))
9694, 95syl5ibrcom 246 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) β†’ (π‘œ = 𝑋 β†’ π‘œ ∈ 𝐹))
9796necon3bd 2955 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) β†’ (Β¬ π‘œ ∈ 𝐹 β†’ π‘œ β‰  𝑋))
9863, 97mpd 15 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) β†’ π‘œ β‰  𝑋)
99 pssdifn0 4366 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((π‘œ βŠ† 𝑋 ∧ π‘œ β‰  𝑋) β†’ (𝑋 βˆ– π‘œ) β‰  βˆ…)
10077, 98, 99syl2anc 585 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) β†’ (𝑋 βˆ– π‘œ) β‰  βˆ…)
101 supfil 23399 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑋 ∈ 𝐽 ∧ (𝑋 βˆ– π‘œ) βŠ† 𝑋 ∧ (𝑋 βˆ– π‘œ) β‰  βˆ…) β†’ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯} ∈ (Filβ€˜π‘‹))
10233, 92, 100, 101syl3anc 1372 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) β†’ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯} ∈ (Filβ€˜π‘‹))
103 filfbas 23352 . . . . . . . . . . . . . . . . . . . . . . . 24 ({π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯} ∈ (Filβ€˜π‘‹) β†’ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯} ∈ (fBasβ€˜π‘‹))
104102, 103syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) β†’ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯} ∈ (fBasβ€˜π‘‹))
105 fbunfip 23373 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐹 ∈ (fBasβ€˜π‘‹) ∧ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯} ∈ (fBasβ€˜π‘‹)) β†’ (Β¬ βˆ… ∈ (fiβ€˜(𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯})) ↔ βˆ€π‘¦ ∈ 𝐹 βˆ€π‘§ ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯} (𝑦 ∩ 𝑧) β‰  βˆ…))
10691, 104, 105syl2anc 585 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) β†’ (Β¬ βˆ… ∈ (fiβ€˜(𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯})) ↔ βˆ€π‘¦ ∈ 𝐹 βˆ€π‘§ ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯} (𝑦 ∩ 𝑧) β‰  βˆ…))
10789, 106mpbird 257 . . . . . . . . . . . . . . . . . . . . 21 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) β†’ Β¬ βˆ… ∈ (fiβ€˜(𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯})))
108 fsubbas 23371 . . . . . . . . . . . . . . . . . . . . . 22 (𝑋 ∈ 𝐹 β†’ ((fiβ€˜(𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯})) ∈ (fBasβ€˜π‘‹) ↔ ((𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯}) βŠ† 𝒫 𝑋 ∧ (𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯}) β‰  βˆ… ∧ Β¬ βˆ… ∈ (fiβ€˜(𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯})))))
10994, 108syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) β†’ ((fiβ€˜(𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯})) ∈ (fBasβ€˜π‘‹) ↔ ((𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯}) βŠ† 𝒫 𝑋 ∧ (𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯}) β‰  βˆ… ∧ Β¬ βˆ… ∈ (fiβ€˜(𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯})))))
11045, 56, 107, 109mpbir3and 1343 . . . . . . . . . . . . . . . . . . . 20 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) β†’ (fiβ€˜(𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯})) ∈ (fBasβ€˜π‘‹))
111 ssfg 23376 . . . . . . . . . . . . . . . . . . . 20 ((fiβ€˜(𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯})) ∈ (fBasβ€˜π‘‹) β†’ (fiβ€˜(𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯})) βŠ† (𝑋filGen(fiβ€˜(𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯}))))
112110, 111syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) β†’ (fiβ€˜(𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯})) βŠ† (𝑋filGen(fiβ€˜(𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯}))))
11340, 112sstrd 3993 . . . . . . . . . . . . . . . . . 18 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) β†’ (𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯}) βŠ† (𝑋filGen(fiβ€˜(𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯}))))
114113unssad 4188 . . . . . . . . . . . . . . . . 17 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) β†’ 𝐹 βŠ† (𝑋filGen(fiβ€˜(𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯}))))
115 fgcl 23382 . . . . . . . . . . . . . . . . . . 19 ((fiβ€˜(𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯})) ∈ (fBasβ€˜π‘‹) β†’ (𝑋filGen(fiβ€˜(𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯}))) ∈ (Filβ€˜π‘‹))
116110, 115syl 17 . . . . . . . . . . . . . . . . . 18 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) β†’ (𝑋filGen(fiβ€˜(𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯}))) ∈ (Filβ€˜π‘‹))
117 sseq2 4009 . . . . . . . . . . . . . . . . . . . 20 (𝑔 = (𝑋filGen(fiβ€˜(𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯}))) β†’ (𝐹 βŠ† 𝑔 ↔ 𝐹 βŠ† (𝑋filGen(fiβ€˜(𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯})))))
118 oveq2 7417 . . . . . . . . . . . . . . . . . . . . 21 (𝑔 = (𝑋filGen(fiβ€˜(𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯}))) β†’ (𝐽 fClus 𝑔) = (𝐽 fClus (𝑋filGen(fiβ€˜(𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯})))))
119118eleq2d 2820 . . . . . . . . . . . . . . . . . . . 20 (𝑔 = (𝑋filGen(fiβ€˜(𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯}))) β†’ (𝐴 ∈ (𝐽 fClus 𝑔) ↔ 𝐴 ∈ (𝐽 fClus (𝑋filGen(fiβ€˜(𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯}))))))
120117, 119imbi12d 345 . . . . . . . . . . . . . . . . . . 19 (𝑔 = (𝑋filGen(fiβ€˜(𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯}))) β†’ ((𝐹 βŠ† 𝑔 β†’ 𝐴 ∈ (𝐽 fClus 𝑔)) ↔ (𝐹 βŠ† (𝑋filGen(fiβ€˜(𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯}))) β†’ 𝐴 ∈ (𝐽 fClus (𝑋filGen(fiβ€˜(𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯})))))))
121120rspcv 3609 . . . . . . . . . . . . . . . . . 18 ((𝑋filGen(fiβ€˜(𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯}))) ∈ (Filβ€˜π‘‹) β†’ (βˆ€π‘” ∈ (Filβ€˜π‘‹)(𝐹 βŠ† 𝑔 β†’ 𝐴 ∈ (𝐽 fClus 𝑔)) β†’ (𝐹 βŠ† (𝑋filGen(fiβ€˜(𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯}))) β†’ 𝐴 ∈ (𝐽 fClus (𝑋filGen(fiβ€˜(𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯})))))))
122116, 121syl 17 . . . . . . . . . . . . . . . . 17 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) β†’ (βˆ€π‘” ∈ (Filβ€˜π‘‹)(𝐹 βŠ† 𝑔 β†’ 𝐴 ∈ (𝐽 fClus 𝑔)) β†’ (𝐹 βŠ† (𝑋filGen(fiβ€˜(𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯}))) β†’ 𝐴 ∈ (𝐽 fClus (𝑋filGen(fiβ€˜(𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯})))))))
123114, 122mpid 44 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) β†’ (βˆ€π‘” ∈ (Filβ€˜π‘‹)(𝐹 βŠ† 𝑔 β†’ 𝐴 ∈ (𝐽 fClus 𝑔)) β†’ 𝐴 ∈ (𝐽 fClus (𝑋filGen(fiβ€˜(𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯}))))))
124 simpr 486 . . . . . . . . . . . . . . . . . 18 ((((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) ∧ 𝐴 ∈ (𝐽 fClus (𝑋filGen(fiβ€˜(𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯}))))) β†’ 𝐴 ∈ (𝐽 fClus (𝑋filGen(fiβ€˜(𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯})))))
125 simplrl 776 . . . . . . . . . . . . . . . . . 18 ((((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) ∧ 𝐴 ∈ (𝐽 fClus (𝑋filGen(fiβ€˜(𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯}))))) β†’ π‘œ ∈ 𝐽)
126 simprrl 780 . . . . . . . . . . . . . . . . . . 19 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) β†’ 𝐴 ∈ π‘œ)
127126adantr 482 . . . . . . . . . . . . . . . . . 18 ((((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) ∧ 𝐴 ∈ (𝐽 fClus (𝑋filGen(fiβ€˜(𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯}))))) β†’ 𝐴 ∈ π‘œ)
128113, 55sseldd 3984 . . . . . . . . . . . . . . . . . . 19 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) β†’ (𝑋 βˆ– π‘œ) ∈ (𝑋filGen(fiβ€˜(𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯}))))
129128adantr 482 . . . . . . . . . . . . . . . . . 18 ((((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) ∧ 𝐴 ∈ (𝐽 fClus (𝑋filGen(fiβ€˜(𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯}))))) β†’ (𝑋 βˆ– π‘œ) ∈ (𝑋filGen(fiβ€˜(𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯}))))
130 fclsopni 23519 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ (𝐽 fClus (𝑋filGen(fiβ€˜(𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯})))) ∧ (π‘œ ∈ 𝐽 ∧ 𝐴 ∈ π‘œ ∧ (𝑋 βˆ– π‘œ) ∈ (𝑋filGen(fiβ€˜(𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯}))))) β†’ (π‘œ ∩ (𝑋 βˆ– π‘œ)) β‰  βˆ…)
131124, 125, 127, 129, 130syl13anc 1373 . . . . . . . . . . . . . . . . 17 ((((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) ∧ 𝐴 ∈ (𝐽 fClus (𝑋filGen(fiβ€˜(𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯}))))) β†’ (π‘œ ∩ (𝑋 βˆ– π‘œ)) β‰  βˆ…)
132131ex 414 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) β†’ (𝐴 ∈ (𝐽 fClus (𝑋filGen(fiβ€˜(𝐹 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘œ) βŠ† π‘₯})))) β†’ (π‘œ ∩ (𝑋 βˆ– π‘œ)) β‰  βˆ…))
133123, 132syld 47 . . . . . . . . . . . . . . 15 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) β†’ (βˆ€π‘” ∈ (Filβ€˜π‘‹)(𝐹 βŠ† 𝑔 β†’ 𝐴 ∈ (𝐽 fClus 𝑔)) β†’ (π‘œ ∩ (𝑋 βˆ– π‘œ)) β‰  βˆ…))
134133necon2bd 2957 . . . . . . . . . . . . . 14 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) β†’ ((π‘œ ∩ (𝑋 βˆ– π‘œ)) = βˆ… β†’ Β¬ βˆ€π‘” ∈ (Filβ€˜π‘‹)(𝐹 βŠ† 𝑔 β†’ 𝐴 ∈ (𝐽 fClus 𝑔))))
13529, 134mpi 20 . . . . . . . . . . . . 13 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (π‘œ ∈ 𝐽 ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹))) β†’ Β¬ βˆ€π‘” ∈ (Filβ€˜π‘‹)(𝐹 βŠ† 𝑔 β†’ 𝐴 ∈ (𝐽 fClus 𝑔)))
136135anassrs 469 . . . . . . . . . . . 12 ((((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ π‘œ ∈ 𝐽) ∧ (𝐴 ∈ π‘œ ∧ Β¬ π‘œ ∈ 𝐹)) β†’ Β¬ βˆ€π‘” ∈ (Filβ€˜π‘‹)(𝐹 βŠ† 𝑔 β†’ 𝐴 ∈ (𝐽 fClus 𝑔)))
137136expr 458 . . . . . . . . . . 11 ((((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ π‘œ ∈ 𝐽) ∧ 𝐴 ∈ π‘œ) β†’ (Β¬ π‘œ ∈ 𝐹 β†’ Β¬ βˆ€π‘” ∈ (Filβ€˜π‘‹)(𝐹 βŠ† 𝑔 β†’ 𝐴 ∈ (𝐽 fClus 𝑔))))
138137con4d 115 . . . . . . . . . 10 ((((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ π‘œ ∈ 𝐽) ∧ 𝐴 ∈ π‘œ) β†’ (βˆ€π‘” ∈ (Filβ€˜π‘‹)(𝐹 βŠ† 𝑔 β†’ 𝐴 ∈ (𝐽 fClus 𝑔)) β†’ π‘œ ∈ 𝐹))
139138ex 414 . . . . . . . . 9 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ π‘œ ∈ 𝐽) β†’ (𝐴 ∈ π‘œ β†’ (βˆ€π‘” ∈ (Filβ€˜π‘‹)(𝐹 βŠ† 𝑔 β†’ 𝐴 ∈ (𝐽 fClus 𝑔)) β†’ π‘œ ∈ 𝐹)))
140139com23 86 . . . . . . . 8 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ π‘œ ∈ 𝐽) β†’ (βˆ€π‘” ∈ (Filβ€˜π‘‹)(𝐹 βŠ† 𝑔 β†’ 𝐴 ∈ (𝐽 fClus 𝑔)) β†’ (𝐴 ∈ π‘œ β†’ π‘œ ∈ 𝐹)))
141140ralrimdva 3155 . . . . . . 7 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) β†’ (βˆ€π‘” ∈ (Filβ€˜π‘‹)(𝐹 βŠ† 𝑔 β†’ 𝐴 ∈ (𝐽 fClus 𝑔)) β†’ βˆ€π‘œ ∈ 𝐽 (𝐴 ∈ π‘œ β†’ π‘œ ∈ 𝐹)))
142 simprr 772 . . . . . . 7 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) β†’ 𝐴 ∈ 𝑋)
143141, 142jctild 527 . . . . . 6 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) β†’ (βˆ€π‘” ∈ (Filβ€˜π‘‹)(𝐹 βŠ† 𝑔 β†’ 𝐴 ∈ (𝐽 fClus 𝑔)) β†’ (𝐴 ∈ 𝑋 ∧ βˆ€π‘œ ∈ 𝐽 (𝐴 ∈ π‘œ β†’ π‘œ ∈ 𝐹))))
144 simprl 770 . . . . . . . 8 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) β†’ 𝐽 ∈ Top)
145144, 4sylib 217 . . . . . . 7 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
146 simpl 484 . . . . . . 7 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) β†’ 𝐹 ∈ (Filβ€˜π‘‹))
147 flimopn 23479 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ βˆ€π‘œ ∈ 𝐽 (𝐴 ∈ π‘œ β†’ π‘œ ∈ 𝐹))))
148145, 146, 147syl2anc 585 . . . . . 6 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) β†’ (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ βˆ€π‘œ ∈ 𝐽 (𝐴 ∈ π‘œ β†’ π‘œ ∈ 𝐹))))
149143, 148sylibrd 259 . . . . 5 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) β†’ (βˆ€π‘” ∈ (Filβ€˜π‘‹)(𝐹 βŠ† 𝑔 β†’ 𝐴 ∈ (𝐽 fClus 𝑔)) β†’ 𝐴 ∈ (𝐽 fLim 𝐹)))
150149ex 414 . . . 4 (𝐹 ∈ (Filβ€˜π‘‹) β†’ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋) β†’ (βˆ€π‘” ∈ (Filβ€˜π‘‹)(𝐹 βŠ† 𝑔 β†’ 𝐴 ∈ (𝐽 fClus 𝑔)) β†’ 𝐴 ∈ (𝐽 fLim 𝐹))))
151150com23 86 . . 3 (𝐹 ∈ (Filβ€˜π‘‹) β†’ (βˆ€π‘” ∈ (Filβ€˜π‘‹)(𝐹 βŠ† 𝑔 β†’ 𝐴 ∈ (𝐽 fClus 𝑔)) β†’ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋) β†’ 𝐴 ∈ (𝐽 fLim 𝐹))))
15228, 151mpdd 43 . 2 (𝐹 ∈ (Filβ€˜π‘‹) β†’ (βˆ€π‘” ∈ (Filβ€˜π‘‹)(𝐹 βŠ† 𝑔 β†’ 𝐴 ∈ (𝐽 fClus 𝑔)) β†’ 𝐴 ∈ (𝐽 fLim 𝐹)))
15315, 152impbid2 225 1 (𝐹 ∈ (Filβ€˜π‘‹) β†’ (𝐴 ∈ (𝐽 fLim 𝐹) ↔ βˆ€π‘” ∈ (Filβ€˜π‘‹)(𝐹 βŠ† 𝑔 β†’ 𝐴 ∈ (𝐽 fClus 𝑔))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062  {crab 3433  Vcvv 3475   βˆ– cdif 3946   βˆͺ cun 3947   ∩ cin 3948   βŠ† wss 3949  βˆ…c0 4323  π’« cpw 4603  βˆͺ cuni 4909  β€˜cfv 6544  (class class class)co 7409  ficfi 9405  fBascfbas 20932  filGencfg 20933  Topctop 22395  TopOnctopon 22412  Filcfil 23349   fLim cflim 23438   fClus cfcls 23440
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1o 8466  df-er 8703  df-en 8940  df-fin 8943  df-fi 9406  df-fbas 20941  df-fg 20942  df-top 22396  df-topon 22413  df-cld 22523  df-ntr 22524  df-cls 22525  df-nei 22602  df-fil 23350  df-flim 23443  df-fcls 23445
This theorem is referenced by:  cnpfcf  23545
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