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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 22611 | . 2 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))) |
3 | 2 | simp2bi 1142 | 1 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐹 ∈ (Fil‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∪ cuni 4831 ‘cfv 6349 (class class class)co 7150 Topctop 21495 clsccl 21620 Filcfil 22447 fClus cfcls 22538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-int 4869 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-fbas 20536 df-fil 22448 df-fcls 22543 |
This theorem is referenced by: fclstopon 22614 fclsopni 22617 fclselbas 22618 fclsfnflim 22629 cnpfcfi 22642 |
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