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Mirrors > Home > MPE Home > Th. List > Mathboxes > opnneieqv | Structured version Visualization version GIF version |
Description: The equivalence between neighborhood and open neighborhood. See opnneieqvv 48593 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 48591 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓 → ∃𝑥 ∈ 𝐽 (𝑆 ⊆ 𝑥 ∧ 𝜓))) |
5 | 1 | opnneir 48588 | . 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 2108 ∃wrex 3076 ⊆ wss 3976 ‘cfv 6575 Topctop 22922 neicnei 23128 |
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 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 |
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 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-top 22923 df-nei 23129 |
This theorem is referenced by: opnneieqvv 48593 sepnsepolem2 48604 sepnsepo 48605 |
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