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Mirrors > Home > MPE Home > Th. List > fclsfil | Structured version Visualization version GIF version |
Description: Reverse closure for the cluster point predicate. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.) |
Ref | Expression |
---|---|
fclsval.x | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
fclsfil | ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐹 ∈ (Fil‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fclsval.x | . . 3 ⊢ 𝑋 = ∪ 𝐽 | |
2 | 1 | isfcls 23358 | . 2 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))) |
3 | 2 | simp2bi 1146 | 1 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐹 ∈ (Fil‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ∀wral 3064 ∪ cuni 4865 ‘cfv 6496 (class class class)co 7356 Topctop 22240 clsccl 22367 Filcfil 23194 fClus cfcls 23285 |
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 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-fv 6504 df-ov 7359 df-oprab 7360 df-mpo 7361 df-fbas 20791 df-fil 23195 df-fcls 23290 |
This theorem is referenced by: fclstopon 23361 fclsopni 23364 fclselbas 23365 fclsfnflim 23376 cnpfcfi 23389 |
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