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Mirrors > Home > MPE Home > Th. List > flimtop | Structured version Visualization version GIF version |
Description: Reverse closure for the limit point predicate. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 9-Aug-2015.) |
Ref | Expression |
---|---|
flimtop | ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | elflim2 21768 | . . 3 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil ∧ 𝐹 ⊆ 𝒫 ∪ 𝐽) ∧ (𝐴 ∈ ∪ 𝐽 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹))) |
3 | 2 | simplbi 476 | . 2 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil ∧ 𝐹 ⊆ 𝒫 ∪ 𝐽)) |
4 | 3 | simp1d 1073 | 1 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 ∈ wcel 1990 ⊆ wss 3574 𝒫 cpw 4158 {csn 4177 ∪ cuni 4436 ran crn 5115 ‘cfv 5888 (class class class)co 6650 Topctop 20698 neicnei 20901 Filcfil 21649 fLim cflim 21738 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-top 20699 df-flim 21743 |
This theorem is referenced by: flimfil 21773 flimtopon 21774 flimss1 21777 flimclsi 21782 hausflimlem 21783 flimsncls 21790 cnpflfi 21803 flimfcls 21830 flimfnfcls 21832 |
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