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Mirrors > Home > MPE Home > Th. List > isnei | Structured version Visualization version GIF version |
Description: The predicate "the class 𝑁 is a neighborhood of 𝑆". (Contributed by FL, 25-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
Ref | Expression |
---|---|
neifval.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
isnei | ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neifval.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐽 | |
2 | 1 | neival 21704 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((nei‘𝐽)‘𝑆) = {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣)}) |
3 | 2 | eleq2d 2898 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ 𝑁 ∈ {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣)})) |
4 | sseq2 3993 | . . . . . . 7 ⊢ (𝑣 = 𝑁 → (𝑔 ⊆ 𝑣 ↔ 𝑔 ⊆ 𝑁)) | |
5 | 4 | anbi2d 630 | . . . . . 6 ⊢ (𝑣 = 𝑁 → ((𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣) ↔ (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁))) |
6 | 5 | rexbidv 3297 | . . . . 5 ⊢ (𝑣 = 𝑁 → (∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣) ↔ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁))) |
7 | 6 | elrab 3680 | . . . 4 ⊢ (𝑁 ∈ {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣)} ↔ (𝑁 ∈ 𝒫 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁))) |
8 | 1 | topopn 21508 | . . . . . 6 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
9 | elpw2g 5240 | . . . . . 6 ⊢ (𝑋 ∈ 𝐽 → (𝑁 ∈ 𝒫 𝑋 ↔ 𝑁 ⊆ 𝑋)) | |
10 | 8, 9 | syl 17 | . . . . 5 ⊢ (𝐽 ∈ Top → (𝑁 ∈ 𝒫 𝑋 ↔ 𝑁 ⊆ 𝑋)) |
11 | 10 | anbi1d 631 | . . . 4 ⊢ (𝐽 ∈ Top → ((𝑁 ∈ 𝒫 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁)) ↔ (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁)))) |
12 | 7, 11 | syl5bb 285 | . . 3 ⊢ (𝐽 ∈ Top → (𝑁 ∈ {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣)} ↔ (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁)))) |
13 | 12 | adantr 483 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑁 ∈ {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣)} ↔ (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁)))) |
14 | 3, 13 | bitrd 281 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∃wrex 3139 {crab 3142 ⊆ wss 3936 𝒫 cpw 4539 ∪ cuni 4832 ‘cfv 6350 Topctop 21495 neicnei 21699 |
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 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 |
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-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-top 21496 df-nei 21700 |
This theorem is referenced by: neiint 21706 isneip 21707 neii1 21708 neii2 21710 neiss 21711 neips 21715 opnneissb 21716 opnssneib 21717 ssnei2 21718 innei 21727 neitr 21782 neitx 22209 neifg 33714 islptre 41892 |
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