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Mirrors > Home > MPE Home > Th. List > fclselbas | Structured version Visualization version GIF version |
Description: A cluster point is in the base set. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.) |
Ref | Expression |
---|---|
fclselbas.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
fclselbas | ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐴 ∈ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fclselbas.1 | . . . . . 6 ⊢ 𝑋 = ∪ 𝐽 | |
2 | 1 | fclsfil 22620 | . . . . 5 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐹 ∈ (Fil‘𝑋)) |
3 | fclstopon 22622 | . . . . 5 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐹 ∈ (Fil‘𝑋))) | |
4 | 2, 3 | mpbird 259 | . . . 4 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐽 ∈ (TopOn‘𝑋)) |
5 | fclsopn 22624 | . . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅)))) | |
6 | 4, 2, 5 | syl2anc 586 | . . 3 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅)))) |
7 | 6 | ibi 269 | . 2 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅))) |
8 | 7 | simpld 497 | 1 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐴 ∈ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 ∩ cin 3937 ∅c0 4293 ∪ cuni 4840 ‘cfv 6357 (class class class)co 7158 TopOnctopon 21520 Filcfil 22455 fClus cfcls 22546 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-fbas 20544 df-top 21504 df-topon 21521 df-cld 21629 df-ntr 21630 df-cls 21631 df-fil 22456 df-fcls 22551 |
This theorem is referenced by: fclsneii 22627 fclsnei 22629 fclsfnflim 22637 flimfnfcls 22638 fcfelbas 22646 cnfcf 22652 cfilfcls 23879 |
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