![]() |
Mathbox for Zhi Wang |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > opnneieqv | Structured version Visualization version GIF version |
Description: The equivalence between neighborhood and open neighborhood. See opnneieqvv 48510 for different dummy variables. (Contributed by Zhi Wang, 31-Aug-2024.) |
Ref | Expression |
---|---|
opnneir.1 | ⊢ (𝜑 → 𝐽 ∈ Top) |
opnneilv.2 | ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝑥) → (𝜓 → 𝜒)) |
opnneil.3 | ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
opnneieqv | ⊢ (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓 ↔ ∃𝑥 ∈ 𝐽 (𝑆 ⊆ 𝑥 ∧ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opnneir.1 | . . 3 ⊢ (𝜑 → 𝐽 ∈ Top) | |
2 | opnneilv.2 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝑥) → (𝜓 → 𝜒)) | |
3 | opnneil.3 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) | |
4 | 1, 2, 3 | opnneil 48508 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓 → ∃𝑥 ∈ 𝐽 (𝑆 ⊆ 𝑥 ∧ 𝜓))) |
5 | 1 | opnneir 48505 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐽 (𝑆 ⊆ 𝑥 ∧ 𝜓) → ∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓)) |
6 | 4, 5 | impbid 212 | 1 ⊢ (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓 ↔ ∃𝑥 ∈ 𝐽 (𝑆 ⊆ 𝑥 ∧ 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2103 ∃wrex 3072 ⊆ wss 3970 ‘cfv 6572 Topctop 22913 neicnei 23119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-ral 3064 df-rex 3073 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-id 5597 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-top 22914 df-nei 23120 |
This theorem is referenced by: opnneieqvv 48510 sepnsepolem2 48521 sepnsepo 48522 |
Copyright terms: Public domain | W3C validator |