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Mirrors > Home > MPE Home > Th. List > fcfneii | Structured version Visualization version GIF version |
Description: A neighborhood of a cluster point of a function contains a function value from every tail. (Contributed by Jeff Hankins, 27-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.) |
Ref | Expression |
---|---|
fcfneii | ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐿)) → (𝑁 ∩ (𝐹 “ 𝑆)) ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fcfnei 22637 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐿 (𝑛 ∩ (𝐹 “ 𝑠)) ≠ ∅))) | |
2 | ineq1 4180 | . . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑛 ∩ (𝐹 “ 𝑠)) = (𝑁 ∩ (𝐹 “ 𝑠))) | |
3 | 2 | neeq1d 3075 | . . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝑛 ∩ (𝐹 “ 𝑠)) ≠ ∅ ↔ (𝑁 ∩ (𝐹 “ 𝑠)) ≠ ∅)) |
4 | imaeq2 5919 | . . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (𝐹 “ 𝑠) = (𝐹 “ 𝑆)) | |
5 | 4 | ineq2d 4188 | . . . . . . . 8 ⊢ (𝑠 = 𝑆 → (𝑁 ∩ (𝐹 “ 𝑠)) = (𝑁 ∩ (𝐹 “ 𝑆))) |
6 | 5 | neeq1d 3075 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → ((𝑁 ∩ (𝐹 “ 𝑠)) ≠ ∅ ↔ (𝑁 ∩ (𝐹 “ 𝑆)) ≠ ∅)) |
7 | 3, 6 | rspc2v 3632 | . . . . . 6 ⊢ ((𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐿) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐿 (𝑛 ∩ (𝐹 “ 𝑠)) ≠ ∅ → (𝑁 ∩ (𝐹 “ 𝑆)) ≠ ∅)) |
8 | 7 | ex 415 | . . . . 5 ⊢ (𝑁 ∈ ((nei‘𝐽)‘{𝐴}) → (𝑆 ∈ 𝐿 → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐿 (𝑛 ∩ (𝐹 “ 𝑠)) ≠ ∅ → (𝑁 ∩ (𝐹 “ 𝑆)) ≠ ∅))) |
9 | 8 | com3r 87 | . . . 4 ⊢ (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐿 (𝑛 ∩ (𝐹 “ 𝑠)) ≠ ∅ → (𝑁 ∈ ((nei‘𝐽)‘{𝐴}) → (𝑆 ∈ 𝐿 → (𝑁 ∩ (𝐹 “ 𝑆)) ≠ ∅))) |
10 | 9 | adantl 484 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐿 (𝑛 ∩ (𝐹 “ 𝑠)) ≠ ∅) → (𝑁 ∈ ((nei‘𝐽)‘{𝐴}) → (𝑆 ∈ 𝐿 → (𝑁 ∩ (𝐹 “ 𝑆)) ≠ ∅))) |
11 | 1, 10 | syl6bi 255 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) → (𝑁 ∈ ((nei‘𝐽)‘{𝐴}) → (𝑆 ∈ 𝐿 → (𝑁 ∩ (𝐹 “ 𝑆)) ≠ ∅)))) |
12 | 11 | 3imp2 1345 | 1 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐿)) → (𝑁 ∩ (𝐹 “ 𝑆)) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 ∩ cin 3934 ∅c0 4290 {csn 4560 “ cima 5552 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 TopOnctopon 21512 neicnei 21699 Filcfil 22447 fClusf cfcf 22539 |
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-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
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-reu 3145 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-iun 4913 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-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-map 8402 df-fbas 20536 df-fg 20537 df-top 21496 df-topon 21513 df-cld 21621 df-ntr 21622 df-cls 21623 df-nei 21700 df-fil 22448 df-fm 22540 df-fcls 22543 df-fcf 22544 |
This theorem is referenced by: (None) |
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