Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > flimelbas | Structured version Visualization version GIF version |
Description: A limit point of a filter belongs to its base set. (Contributed by Jeff Hankins, 4-Sep-2009.) (Revised by Mario Carneiro, 9-Apr-2015.) |
Ref | Expression |
---|---|
flimuni.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
flimelbas | ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐴 ∈ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | flimuni.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐽 | |
2 | 1 | elflim2 22815 | . . 3 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil ∧ 𝐹 ⊆ 𝒫 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹))) |
3 | 2 | simprbi 500 | . 2 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)) |
4 | 3 | simpld 498 | 1 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐴 ∈ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ⊆ wss 3853 𝒫 cpw 4499 {csn 4527 ∪ cuni 4805 ran crn 5537 ‘cfv 6358 (class class class)co 7191 Topctop 21744 neicnei 21948 Filcfil 22696 fLim cflim 22785 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-iota 6316 df-fun 6360 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-top 21745 df-flim 22790 |
This theorem is referenced by: flimfil 22820 flimss2 22823 flimss1 22824 flimclsi 22829 hausflimi 22831 flimsncls 22837 cnpflfi 22850 cnflf 22853 cnflf2 22854 flimcfil 24165 |
Copyright terms: Public domain | W3C validator |