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Theorem fclscf 23520
Description: Characterization of fineness of topologies in terms of cluster points. (Contributed by Jeff Hankins, 12-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
Assertion
Ref Expression
fclscf ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) β†’ (𝐽 βŠ† 𝐾 ↔ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓)))
Distinct variable groups:   𝑓,𝐽   𝑓,𝐾   𝑓,𝑋
Proof of Theorem fclscf
Dummy variables 𝑛 𝑒 π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll 765 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ (𝐽 βŠ† 𝐾 ∧ π‘₯ ∈ (𝐾 fClus 𝑓))) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
2 simplr 767 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ (𝐽 βŠ† 𝐾 ∧ π‘₯ ∈ (𝐾 fClus 𝑓))) β†’ 𝐾 ∈ (TopOnβ€˜π‘‹))
3 fclstopon 23507 . . . . . . . . 9 (π‘₯ ∈ (𝐾 fClus 𝑓) β†’ (𝐾 ∈ (TopOnβ€˜π‘‹) ↔ 𝑓 ∈ (Filβ€˜π‘‹)))
43ad2antll 727 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ (𝐽 βŠ† 𝐾 ∧ π‘₯ ∈ (𝐾 fClus 𝑓))) β†’ (𝐾 ∈ (TopOnβ€˜π‘‹) ↔ 𝑓 ∈ (Filβ€˜π‘‹)))
52, 4mpbid 231 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ (𝐽 βŠ† 𝐾 ∧ π‘₯ ∈ (𝐾 fClus 𝑓))) β†’ 𝑓 ∈ (Filβ€˜π‘‹))
6 simprl 769 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ (𝐽 βŠ† 𝐾 ∧ π‘₯ ∈ (𝐾 fClus 𝑓))) β†’ 𝐽 βŠ† 𝐾)
7 fclsss1 23517 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑓 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) β†’ (𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓))
81, 5, 6, 7syl3anc 1371 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ (𝐽 βŠ† 𝐾 ∧ π‘₯ ∈ (𝐾 fClus 𝑓))) β†’ (𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓))
9 simprr 771 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ (𝐽 βŠ† 𝐾 ∧ π‘₯ ∈ (𝐾 fClus 𝑓))) β†’ π‘₯ ∈ (𝐾 fClus 𝑓))
108, 9sseldd 3982 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ (𝐽 βŠ† 𝐾 ∧ π‘₯ ∈ (𝐾 fClus 𝑓))) β†’ π‘₯ ∈ (𝐽 fClus 𝑓))
1110expr 457 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ 𝐽 βŠ† 𝐾) β†’ (π‘₯ ∈ (𝐾 fClus 𝑓) β†’ π‘₯ ∈ (𝐽 fClus 𝑓)))
1211ssrdv 3987 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ 𝐽 βŠ† 𝐾) β†’ (𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓))
1312ralrimivw 3150 . 2 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ 𝐽 βŠ† 𝐾) β†’ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓))
14 simpllr 774 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓)) ∧ π‘₯ ∈ 𝐽) β†’ 𝐾 ∈ (TopOnβ€˜π‘‹))
15 toponmax 22419 . . . . . . . . 9 (𝐾 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 ∈ 𝐾)
16 ssid 4003 . . . . . . . . . . 11 𝑋 βŠ† 𝑋
17 eleq2 2822 . . . . . . . . . . . . 13 (𝑒 = 𝑋 β†’ (𝑦 ∈ 𝑒 ↔ 𝑦 ∈ 𝑋))
18 sseq1 4006 . . . . . . . . . . . . 13 (𝑒 = 𝑋 β†’ (𝑒 βŠ† 𝑋 ↔ 𝑋 βŠ† 𝑋))
1917, 18anbi12d 631 . . . . . . . . . . . 12 (𝑒 = 𝑋 β†’ ((𝑦 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑋) ↔ (𝑦 ∈ 𝑋 ∧ 𝑋 βŠ† 𝑋)))
2019rspcev 3612 . . . . . . . . . . 11 ((𝑋 ∈ 𝐾 ∧ (𝑦 ∈ 𝑋 ∧ 𝑋 βŠ† 𝑋)) β†’ βˆƒπ‘’ ∈ 𝐾 (𝑦 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑋))
2116, 20mpanr2 702 . . . . . . . . . 10 ((𝑋 ∈ 𝐾 ∧ 𝑦 ∈ 𝑋) β†’ βˆƒπ‘’ ∈ 𝐾 (𝑦 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑋))
2221ex 413 . . . . . . . . 9 (𝑋 ∈ 𝐾 β†’ (𝑦 ∈ 𝑋 β†’ βˆƒπ‘’ ∈ 𝐾 (𝑦 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑋)))
2314, 15, 223syl 18 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓)) ∧ π‘₯ ∈ 𝐽) β†’ (𝑦 ∈ 𝑋 β†’ βˆƒπ‘’ ∈ 𝐾 (𝑦 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑋)))
24 eleq2 2822 . . . . . . . . 9 (π‘₯ = 𝑋 β†’ (𝑦 ∈ π‘₯ ↔ 𝑦 ∈ 𝑋))
25 sseq2 4007 . . . . . . . . . . 11 (π‘₯ = 𝑋 β†’ (𝑒 βŠ† π‘₯ ↔ 𝑒 βŠ† 𝑋))
2625anbi2d 629 . . . . . . . . . 10 (π‘₯ = 𝑋 β†’ ((𝑦 ∈ 𝑒 ∧ 𝑒 βŠ† π‘₯) ↔ (𝑦 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑋)))
2726rexbidv 3178 . . . . . . . . 9 (π‘₯ = 𝑋 β†’ (βˆƒπ‘’ ∈ 𝐾 (𝑦 ∈ 𝑒 ∧ 𝑒 βŠ† π‘₯) ↔ βˆƒπ‘’ ∈ 𝐾 (𝑦 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑋)))
2824, 27imbi12d 344 . . . . . . . 8 (π‘₯ = 𝑋 β†’ ((𝑦 ∈ π‘₯ β†’ βˆƒπ‘’ ∈ 𝐾 (𝑦 ∈ 𝑒 ∧ 𝑒 βŠ† π‘₯)) ↔ (𝑦 ∈ 𝑋 β†’ βˆƒπ‘’ ∈ 𝐾 (𝑦 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑋))))
2923, 28syl5ibrcom 246 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓)) ∧ π‘₯ ∈ 𝐽) β†’ (π‘₯ = 𝑋 β†’ (𝑦 ∈ π‘₯ β†’ βˆƒπ‘’ ∈ 𝐾 (𝑦 ∈ 𝑒 ∧ 𝑒 βŠ† π‘₯))))
30 simplll 773 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓)) ∧ (π‘₯ ∈ 𝐽 ∧ (π‘₯ β‰  𝑋 ∧ 𝑦 ∈ π‘₯))) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
31 simprl 769 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓)) ∧ (π‘₯ ∈ 𝐽 ∧ (π‘₯ β‰  𝑋 ∧ 𝑦 ∈ π‘₯))) β†’ π‘₯ ∈ 𝐽)
32 simprrr 780 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓)) ∧ (π‘₯ ∈ 𝐽 ∧ (π‘₯ β‰  𝑋 ∧ 𝑦 ∈ π‘₯))) β†’ 𝑦 ∈ π‘₯)
33 supnfcls 23515 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) β†’ Β¬ 𝑦 ∈ (𝐽 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘₯) βŠ† 𝑦}))
3430, 31, 32, 33syl3anc 1371 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓)) ∧ (π‘₯ ∈ 𝐽 ∧ (π‘₯ β‰  𝑋 ∧ 𝑦 ∈ π‘₯))) β†’ Β¬ 𝑦 ∈ (𝐽 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘₯) βŠ† 𝑦}))
35 toponss 22420 . . . . . . . . . . . . . . 15 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘₯ ∈ 𝐽) β†’ π‘₯ βŠ† 𝑋)
3630, 31, 35syl2anc 584 . . . . . . . . . . . . . 14 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓)) ∧ (π‘₯ ∈ 𝐽 ∧ (π‘₯ β‰  𝑋 ∧ 𝑦 ∈ π‘₯))) β†’ π‘₯ βŠ† 𝑋)
3736, 32sseldd 3982 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓)) ∧ (π‘₯ ∈ 𝐽 ∧ (π‘₯ β‰  𝑋 ∧ 𝑦 ∈ π‘₯))) β†’ 𝑦 ∈ 𝑋)
38 simpllr 774 . . . . . . . . . . . . . 14 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓)) ∧ (π‘₯ ∈ 𝐽 ∧ (π‘₯ β‰  𝑋 ∧ 𝑦 ∈ π‘₯))) β†’ 𝐾 ∈ (TopOnβ€˜π‘‹))
39 toponmax 22419 . . . . . . . . . . . . . . . 16 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 ∈ 𝐽)
4030, 39syl 17 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓)) ∧ (π‘₯ ∈ 𝐽 ∧ (π‘₯ β‰  𝑋 ∧ 𝑦 ∈ π‘₯))) β†’ 𝑋 ∈ 𝐽)
41 difssd 4131 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓)) ∧ (π‘₯ ∈ 𝐽 ∧ (π‘₯ β‰  𝑋 ∧ 𝑦 ∈ π‘₯))) β†’ (𝑋 βˆ– π‘₯) βŠ† 𝑋)
42 simprrl 779 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓)) ∧ (π‘₯ ∈ 𝐽 ∧ (π‘₯ β‰  𝑋 ∧ 𝑦 ∈ π‘₯))) β†’ π‘₯ β‰  𝑋)
43 pssdifn0 4364 . . . . . . . . . . . . . . . 16 ((π‘₯ βŠ† 𝑋 ∧ π‘₯ β‰  𝑋) β†’ (𝑋 βˆ– π‘₯) β‰  βˆ…)
4436, 42, 43syl2anc 584 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓)) ∧ (π‘₯ ∈ 𝐽 ∧ (π‘₯ β‰  𝑋 ∧ 𝑦 ∈ π‘₯))) β†’ (𝑋 βˆ– π‘₯) β‰  βˆ…)
45 supfil 23390 . . . . . . . . . . . . . . 15 ((𝑋 ∈ 𝐽 ∧ (𝑋 βˆ– π‘₯) βŠ† 𝑋 ∧ (𝑋 βˆ– π‘₯) β‰  βˆ…) β†’ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘₯) βŠ† 𝑦} ∈ (Filβ€˜π‘‹))
4640, 41, 44, 45syl3anc 1371 . . . . . . . . . . . . . 14 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓)) ∧ (π‘₯ ∈ 𝐽 ∧ (π‘₯ β‰  𝑋 ∧ 𝑦 ∈ π‘₯))) β†’ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘₯) βŠ† 𝑦} ∈ (Filβ€˜π‘‹))
47 fclsopn 23509 . . . . . . . . . . . . . 14 ((𝐾 ∈ (TopOnβ€˜π‘‹) ∧ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘₯) βŠ† 𝑦} ∈ (Filβ€˜π‘‹)) β†’ (𝑦 ∈ (𝐾 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘₯) βŠ† 𝑦}) ↔ (𝑦 ∈ 𝑋 ∧ βˆ€π‘’ ∈ 𝐾 (𝑦 ∈ 𝑒 β†’ βˆ€π‘› ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘₯) βŠ† 𝑦} (𝑒 ∩ 𝑛) β‰  βˆ…))))
4838, 46, 47syl2anc 584 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓)) ∧ (π‘₯ ∈ 𝐽 ∧ (π‘₯ β‰  𝑋 ∧ 𝑦 ∈ π‘₯))) β†’ (𝑦 ∈ (𝐾 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘₯) βŠ† 𝑦}) ↔ (𝑦 ∈ 𝑋 ∧ βˆ€π‘’ ∈ 𝐾 (𝑦 ∈ 𝑒 β†’ βˆ€π‘› ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘₯) βŠ† 𝑦} (𝑒 ∩ 𝑛) β‰  βˆ…))))
4937, 48mpbirand 705 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓)) ∧ (π‘₯ ∈ 𝐽 ∧ (π‘₯ β‰  𝑋 ∧ 𝑦 ∈ π‘₯))) β†’ (𝑦 ∈ (𝐾 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘₯) βŠ† 𝑦}) ↔ βˆ€π‘’ ∈ 𝐾 (𝑦 ∈ 𝑒 β†’ βˆ€π‘› ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘₯) βŠ† 𝑦} (𝑒 ∩ 𝑛) β‰  βˆ…)))
50 oveq2 7413 . . . . . . . . . . . . . . 15 (𝑓 = {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘₯) βŠ† 𝑦} β†’ (𝐾 fClus 𝑓) = (𝐾 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘₯) βŠ† 𝑦}))
51 oveq2 7413 . . . . . . . . . . . . . . 15 (𝑓 = {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘₯) βŠ† 𝑦} β†’ (𝐽 fClus 𝑓) = (𝐽 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘₯) βŠ† 𝑦}))
5250, 51sseq12d 4014 . . . . . . . . . . . . . 14 (𝑓 = {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘₯) βŠ† 𝑦} β†’ ((𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓) ↔ (𝐾 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘₯) βŠ† 𝑦}) βŠ† (𝐽 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘₯) βŠ† 𝑦})))
53 simplr 767 . . . . . . . . . . . . . 14 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓)) ∧ (π‘₯ ∈ 𝐽 ∧ (π‘₯ β‰  𝑋 ∧ 𝑦 ∈ π‘₯))) β†’ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓))
5452, 53, 46rspcdva 3613 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓)) ∧ (π‘₯ ∈ 𝐽 ∧ (π‘₯ β‰  𝑋 ∧ 𝑦 ∈ π‘₯))) β†’ (𝐾 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘₯) βŠ† 𝑦}) βŠ† (𝐽 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘₯) βŠ† 𝑦}))
5554sseld 3980 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓)) ∧ (π‘₯ ∈ 𝐽 ∧ (π‘₯ β‰  𝑋 ∧ 𝑦 ∈ π‘₯))) β†’ (𝑦 ∈ (𝐾 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘₯) βŠ† 𝑦}) β†’ 𝑦 ∈ (𝐽 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘₯) βŠ† 𝑦})))
5649, 55sylbird 259 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓)) ∧ (π‘₯ ∈ 𝐽 ∧ (π‘₯ β‰  𝑋 ∧ 𝑦 ∈ π‘₯))) β†’ (βˆ€π‘’ ∈ 𝐾 (𝑦 ∈ 𝑒 β†’ βˆ€π‘› ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘₯) βŠ† 𝑦} (𝑒 ∩ 𝑛) β‰  βˆ…) β†’ 𝑦 ∈ (𝐽 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘₯) βŠ† 𝑦})))
5734, 56mtod 197 . . . . . . . . . 10 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓)) ∧ (π‘₯ ∈ 𝐽 ∧ (π‘₯ β‰  𝑋 ∧ 𝑦 ∈ π‘₯))) β†’ Β¬ βˆ€π‘’ ∈ 𝐾 (𝑦 ∈ 𝑒 β†’ βˆ€π‘› ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘₯) βŠ† 𝑦} (𝑒 ∩ 𝑛) β‰  βˆ…))
58 rexanali 3102 . . . . . . . . . . 11 (βˆƒπ‘’ ∈ 𝐾 (𝑦 ∈ 𝑒 ∧ Β¬ βˆ€π‘› ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘₯) βŠ† 𝑦} (𝑒 ∩ 𝑛) β‰  βˆ…) ↔ Β¬ βˆ€π‘’ ∈ 𝐾 (𝑦 ∈ 𝑒 β†’ βˆ€π‘› ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘₯) βŠ† 𝑦} (𝑒 ∩ 𝑛) β‰  βˆ…))
59 rexnal 3100 . . . . . . . . . . . . . 14 (βˆƒπ‘› ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘₯) βŠ† 𝑦} Β¬ (𝑒 ∩ 𝑛) β‰  βˆ… ↔ Β¬ βˆ€π‘› ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘₯) βŠ† 𝑦} (𝑒 ∩ 𝑛) β‰  βˆ…)
60 sseq2 4007 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑛 β†’ ((𝑋 βˆ– π‘₯) βŠ† 𝑦 ↔ (𝑋 βˆ– π‘₯) βŠ† 𝑛))
6160elrab 3682 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘₯) βŠ† 𝑦} ↔ (𝑛 ∈ 𝒫 𝑋 ∧ (𝑋 βˆ– π‘₯) βŠ† 𝑛))
62 sslin 4233 . . . . . . . . . . . . . . . . 17 ((𝑋 βˆ– π‘₯) βŠ† 𝑛 β†’ (𝑒 ∩ (𝑋 βˆ– π‘₯)) βŠ† (𝑒 ∩ 𝑛))
6361, 62simplbiim 505 . . . . . . . . . . . . . . . 16 (𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘₯) βŠ† 𝑦} β†’ (𝑒 ∩ (𝑋 βˆ– π‘₯)) βŠ† (𝑒 ∩ 𝑛))
64 ssn0 4399 . . . . . . . . . . . . . . . . . . . 20 (((𝑒 ∩ (𝑋 βˆ– π‘₯)) βŠ† (𝑒 ∩ 𝑛) ∧ (𝑒 ∩ (𝑋 βˆ– π‘₯)) β‰  βˆ…) β†’ (𝑒 ∩ 𝑛) β‰  βˆ…)
6564ex 413 . . . . . . . . . . . . . . . . . . 19 ((𝑒 ∩ (𝑋 βˆ– π‘₯)) βŠ† (𝑒 ∩ 𝑛) β†’ ((𝑒 ∩ (𝑋 βˆ– π‘₯)) β‰  βˆ… β†’ (𝑒 ∩ 𝑛) β‰  βˆ…))
6665necon1bd 2958 . . . . . . . . . . . . . . . . . 18 ((𝑒 ∩ (𝑋 βˆ– π‘₯)) βŠ† (𝑒 ∩ 𝑛) β†’ (Β¬ (𝑒 ∩ 𝑛) β‰  βˆ… β†’ (𝑒 ∩ (𝑋 βˆ– π‘₯)) = βˆ…))
67 inssdif0 4368 . . . . . . . . . . . . . . . . . 18 ((𝑒 ∩ 𝑋) βŠ† π‘₯ ↔ (𝑒 ∩ (𝑋 βˆ– π‘₯)) = βˆ…)
6866, 67syl6ibr 251 . . . . . . . . . . . . . . . . 17 ((𝑒 ∩ (𝑋 βˆ– π‘₯)) βŠ† (𝑒 ∩ 𝑛) β†’ (Β¬ (𝑒 ∩ 𝑛) β‰  βˆ… β†’ (𝑒 ∩ 𝑋) βŠ† π‘₯))
69 toponss 22420 . . . . . . . . . . . . . . . . . . . . 21 ((𝐾 ∈ (TopOnβ€˜π‘‹) ∧ 𝑒 ∈ 𝐾) β†’ 𝑒 βŠ† 𝑋)
7038, 69sylan 580 . . . . . . . . . . . . . . . . . . . 20 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓)) ∧ (π‘₯ ∈ 𝐽 ∧ (π‘₯ β‰  𝑋 ∧ 𝑦 ∈ π‘₯))) ∧ 𝑒 ∈ 𝐾) β†’ 𝑒 βŠ† 𝑋)
71 df-ss 3964 . . . . . . . . . . . . . . . . . . . 20 (𝑒 βŠ† 𝑋 ↔ (𝑒 ∩ 𝑋) = 𝑒)
7270, 71sylib 217 . . . . . . . . . . . . . . . . . . 19 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓)) ∧ (π‘₯ ∈ 𝐽 ∧ (π‘₯ β‰  𝑋 ∧ 𝑦 ∈ π‘₯))) ∧ 𝑒 ∈ 𝐾) β†’ (𝑒 ∩ 𝑋) = 𝑒)
7372sseq1d 4012 . . . . . . . . . . . . . . . . . 18 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓)) ∧ (π‘₯ ∈ 𝐽 ∧ (π‘₯ β‰  𝑋 ∧ 𝑦 ∈ π‘₯))) ∧ 𝑒 ∈ 𝐾) β†’ ((𝑒 ∩ 𝑋) βŠ† π‘₯ ↔ 𝑒 βŠ† π‘₯))
7473biimpd 228 . . . . . . . . . . . . . . . . 17 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓)) ∧ (π‘₯ ∈ 𝐽 ∧ (π‘₯ β‰  𝑋 ∧ 𝑦 ∈ π‘₯))) ∧ 𝑒 ∈ 𝐾) β†’ ((𝑒 ∩ 𝑋) βŠ† π‘₯ β†’ 𝑒 βŠ† π‘₯))
7568, 74syl9r 78 . . . . . . . . . . . . . . . 16 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓)) ∧ (π‘₯ ∈ 𝐽 ∧ (π‘₯ β‰  𝑋 ∧ 𝑦 ∈ π‘₯))) ∧ 𝑒 ∈ 𝐾) β†’ ((𝑒 ∩ (𝑋 βˆ– π‘₯)) βŠ† (𝑒 ∩ 𝑛) β†’ (Β¬ (𝑒 ∩ 𝑛) β‰  βˆ… β†’ 𝑒 βŠ† π‘₯)))
7663, 75syl5 34 . . . . . . . . . . . . . . 15 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓)) ∧ (π‘₯ ∈ 𝐽 ∧ (π‘₯ β‰  𝑋 ∧ 𝑦 ∈ π‘₯))) ∧ 𝑒 ∈ 𝐾) β†’ (𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘₯) βŠ† 𝑦} β†’ (Β¬ (𝑒 ∩ 𝑛) β‰  βˆ… β†’ 𝑒 βŠ† π‘₯)))
7776rexlimdv 3153 . . . . . . . . . . . . . 14 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓)) ∧ (π‘₯ ∈ 𝐽 ∧ (π‘₯ β‰  𝑋 ∧ 𝑦 ∈ π‘₯))) ∧ 𝑒 ∈ 𝐾) β†’ (βˆƒπ‘› ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘₯) βŠ† 𝑦} Β¬ (𝑒 ∩ 𝑛) β‰  βˆ… β†’ 𝑒 βŠ† π‘₯))
7859, 77biimtrrid 242 . . . . . . . . . . . . 13 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓)) ∧ (π‘₯ ∈ 𝐽 ∧ (π‘₯ β‰  𝑋 ∧ 𝑦 ∈ π‘₯))) ∧ 𝑒 ∈ 𝐾) β†’ (Β¬ βˆ€π‘› ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘₯) βŠ† 𝑦} (𝑒 ∩ 𝑛) β‰  βˆ… β†’ 𝑒 βŠ† π‘₯))
7978anim2d 612 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓)) ∧ (π‘₯ ∈ 𝐽 ∧ (π‘₯ β‰  𝑋 ∧ 𝑦 ∈ π‘₯))) ∧ 𝑒 ∈ 𝐾) β†’ ((𝑦 ∈ 𝑒 ∧ Β¬ βˆ€π‘› ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘₯) βŠ† 𝑦} (𝑒 ∩ 𝑛) β‰  βˆ…) β†’ (𝑦 ∈ 𝑒 ∧ 𝑒 βŠ† π‘₯)))
8079reximdva 3168 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓)) ∧ (π‘₯ ∈ 𝐽 ∧ (π‘₯ β‰  𝑋 ∧ 𝑦 ∈ π‘₯))) β†’ (βˆƒπ‘’ ∈ 𝐾 (𝑦 ∈ 𝑒 ∧ Β¬ βˆ€π‘› ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘₯) βŠ† 𝑦} (𝑒 ∩ 𝑛) β‰  βˆ…) β†’ βˆƒπ‘’ ∈ 𝐾 (𝑦 ∈ 𝑒 ∧ 𝑒 βŠ† π‘₯)))
8158, 80biimtrrid 242 . . . . . . . . . 10 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓)) ∧ (π‘₯ ∈ 𝐽 ∧ (π‘₯ β‰  𝑋 ∧ 𝑦 ∈ π‘₯))) β†’ (Β¬ βˆ€π‘’ ∈ 𝐾 (𝑦 ∈ 𝑒 β†’ βˆ€π‘› ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 βˆ– π‘₯) βŠ† 𝑦} (𝑒 ∩ 𝑛) β‰  βˆ…) β†’ βˆƒπ‘’ ∈ 𝐾 (𝑦 ∈ 𝑒 ∧ 𝑒 βŠ† π‘₯)))
8257, 81mpd 15 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓)) ∧ (π‘₯ ∈ 𝐽 ∧ (π‘₯ β‰  𝑋 ∧ 𝑦 ∈ π‘₯))) β†’ βˆƒπ‘’ ∈ 𝐾 (𝑦 ∈ 𝑒 ∧ 𝑒 βŠ† π‘₯))
8382anassrs 468 . . . . . . . 8 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓)) ∧ π‘₯ ∈ 𝐽) ∧ (π‘₯ β‰  𝑋 ∧ 𝑦 ∈ π‘₯)) β†’ βˆƒπ‘’ ∈ 𝐾 (𝑦 ∈ 𝑒 ∧ 𝑒 βŠ† π‘₯))
8483exp32 421 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓)) ∧ π‘₯ ∈ 𝐽) β†’ (π‘₯ β‰  𝑋 β†’ (𝑦 ∈ π‘₯ β†’ βˆƒπ‘’ ∈ 𝐾 (𝑦 ∈ 𝑒 ∧ 𝑒 βŠ† π‘₯))))
8529, 84pm2.61dne 3028 . . . . . 6 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓)) ∧ π‘₯ ∈ 𝐽) β†’ (𝑦 ∈ π‘₯ β†’ βˆƒπ‘’ ∈ 𝐾 (𝑦 ∈ 𝑒 ∧ 𝑒 βŠ† π‘₯)))
8685ralrimiv 3145 . . . . 5 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓)) ∧ π‘₯ ∈ 𝐽) β†’ βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ 𝐾 (𝑦 ∈ 𝑒 ∧ 𝑒 βŠ† π‘₯))
87 topontop 22406 . . . . . 6 (𝐾 ∈ (TopOnβ€˜π‘‹) β†’ 𝐾 ∈ Top)
88 eltop2 22469 . . . . . 6 (𝐾 ∈ Top β†’ (π‘₯ ∈ 𝐾 ↔ βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ 𝐾 (𝑦 ∈ 𝑒 ∧ 𝑒 βŠ† π‘₯)))
8914, 87, 883syl 18 . . . . 5 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓)) ∧ π‘₯ ∈ 𝐽) β†’ (π‘₯ ∈ 𝐾 ↔ βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ 𝐾 (𝑦 ∈ 𝑒 ∧ 𝑒 βŠ† π‘₯)))
9086, 89mpbird 256 . . . 4 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓)) ∧ π‘₯ ∈ 𝐽) β†’ π‘₯ ∈ 𝐾)
9190ex 413 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓)) β†’ (π‘₯ ∈ 𝐽 β†’ π‘₯ ∈ 𝐾))
9291ssrdv 3987 . 2 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓)) β†’ 𝐽 βŠ† 𝐾)
9313, 92impbida 799 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) β†’ (𝐽 βŠ† 𝐾 ↔ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐾 fClus 𝑓) βŠ† (𝐽 fClus 𝑓)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070  {crab 3432   βˆ– cdif 3944   ∩ cin 3946   βŠ† wss 3947  βˆ…c0 4321  π’« cpw 4601  β€˜cfv 6540  (class class class)co 7405  Topctop 22386  TopOnctopon 22403  Filcfil 23340   fClus cfcls 23431
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-topgen 17385  df-fbas 20933  df-top 22387  df-topon 22404  df-cld 22514  df-ntr 22515  df-cls 22516  df-fil 23341  df-fcls 23436
This theorem is referenced by: (None)
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