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| Mirrors > Home > MPE Home > Th. List > neiuni | Structured version Visualization version GIF version | ||
| Description: The union of the neighborhoods of a set equals the topology's underlying set. (Contributed by FL, 18-Sep-2007.) (Revised by Mario Carneiro, 9-Apr-2015.) |
| Ref | Expression |
|---|---|
| tpnei.1 | ⊢ 𝑋 = ∪ 𝐽 |
| Ref | Expression |
|---|---|
| neiuni | ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑋 = ∪ ((nei‘𝐽)‘𝑆)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | tpnei.1 | . . . . 5 ⊢ 𝑋 = ∪ 𝐽 | |
| 2 | 1 | tpnei 23129 | . . . 4 ⊢ (𝐽 ∈ Top → (𝑆 ⊆ 𝑋 ↔ 𝑋 ∈ ((nei‘𝐽)‘𝑆))) |
| 3 | 2 | biimpa 476 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑋 ∈ ((nei‘𝐽)‘𝑆)) |
| 4 | elssuni 4937 | . . 3 ⊢ (𝑋 ∈ ((nei‘𝐽)‘𝑆) → 𝑋 ⊆ ∪ ((nei‘𝐽)‘𝑆)) | |
| 5 | 3, 4 | syl 17 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑋 ⊆ ∪ ((nei‘𝐽)‘𝑆)) |
| 6 | 1 | neii1 23114 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆)) → 𝑥 ⊆ 𝑋) |
| 7 | 6 | ex 412 | . . . . 5 ⊢ (𝐽 ∈ Top → (𝑥 ∈ ((nei‘𝐽)‘𝑆) → 𝑥 ⊆ 𝑋)) |
| 8 | 7 | adantr 480 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑥 ∈ ((nei‘𝐽)‘𝑆) → 𝑥 ⊆ 𝑋)) |
| 9 | 8 | ralrimiv 3145 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∀𝑥 ∈ ((nei‘𝐽)‘𝑆)𝑥 ⊆ 𝑋) |
| 10 | unissb 4939 | . . 3 ⊢ (∪ ((nei‘𝐽)‘𝑆) ⊆ 𝑋 ↔ ∀𝑥 ∈ ((nei‘𝐽)‘𝑆)𝑥 ⊆ 𝑋) | |
| 11 | 9, 10 | sylibr 234 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∪ ((nei‘𝐽)‘𝑆) ⊆ 𝑋) |
| 12 | 5, 11 | eqssd 4001 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑋 = ∪ ((nei‘𝐽)‘𝑆)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ⊆ wss 3951 ∪ cuni 4907 ‘cfv 6561 Topctop 22899 neicnei 23105 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-top 22900 df-nei 23106 |
| This theorem is referenced by: neifil 23888 |
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