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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sclnbgrel | Structured version Visualization version GIF version |
Description: Characterization of a member 𝑋 of the semiclosed neighborhood of a vertex 𝑁 in a graph 𝐺. (Contributed by AV, 16-May-2025.) |
Ref | Expression |
---|---|
dfsclnbgr2.v | ⊢ 𝑉 = (Vtx‘𝐺) |
dfsclnbgr2.s | ⊢ 𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒} |
dfsclnbgr2.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
sclnbgrel | ⊢ (𝑋 ∈ 𝑆 ↔ (𝑋 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑁, 𝑋} ⊆ 𝑒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsclnbgr2.s | . . 3 ⊢ 𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒} | |
2 | 1 | eleq2i 2836 | . 2 ⊢ (𝑋 ∈ 𝑆 ↔ 𝑋 ∈ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}) |
3 | preq2 4759 | . . . . 5 ⊢ (𝑛 = 𝑋 → {𝑁, 𝑛} = {𝑁, 𝑋}) | |
4 | 3 | sseq1d 4040 | . . . 4 ⊢ (𝑛 = 𝑋 → ({𝑁, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑋} ⊆ 𝑒)) |
5 | 4 | rexbidv 3185 | . . 3 ⊢ (𝑛 = 𝑋 → (∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ 𝐸 {𝑁, 𝑋} ⊆ 𝑒)) |
6 | 5 | elrab 3708 | . 2 ⊢ (𝑋 ∈ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒} ↔ (𝑋 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑁, 𝑋} ⊆ 𝑒)) |
7 | 2, 6 | bitri 275 | 1 ⊢ (𝑋 ∈ 𝑆 ↔ (𝑋 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑁, 𝑋} ⊆ 𝑒)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ∃wrex 3076 {crab 3443 ⊆ wss 3976 {cpr 4650 ‘cfv 6573 Vtxcvtx 29031 Edgcedg 29082 |
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-ext 2711 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-tru 1540 df-ex 1778 df-sb 2065 df-clab 2718 df-cleq 2732 df-clel 2819 df-rex 3077 df-rab 3444 df-v 3490 df-un 3981 df-ss 3993 df-sn 4649 df-pr 4651 |
This theorem is referenced by: sclnbgrelself 47720 sclnbgrisvtx 47721 |
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