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Mirrors > Home > MPE Home > Th. List > Mathboxes > vopnbgrel | Structured version Visualization version GIF version |
Description: Characterization of a member 𝑋 of the semiopen neighborhood of a vertex 𝑁 in a graph 𝐺. (Contributed by AV, 16-May-2025.) |
Ref | Expression |
---|---|
dfvopnbgr2.v | ⊢ 𝑉 = (Vtx‘𝐺) |
dfvopnbgr2.e | ⊢ 𝐸 = (Edg‘𝐺) |
dfvopnbgr2.u | ⊢ 𝑈 = {𝑛 ∈ 𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁}))} |
Ref | Expression |
---|---|
vopnbgrel | ⊢ (𝑁 ∈ 𝑉 → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 ((𝑋 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑋 ∈ 𝑒) ∨ (𝑋 = 𝑁 ∧ 𝑒 = {𝑋}))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfvopnbgr2.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | dfvopnbgr2.e | . . . 4 ⊢ 𝐸 = (Edg‘𝐺) | |
3 | dfvopnbgr2.u | . . . 4 ⊢ 𝑈 = {𝑛 ∈ 𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁}))} | |
4 | 1, 2, 3 | dfvopnbgr2 47777 | . . 3 ⊢ (𝑁 ∈ 𝑉 → 𝑈 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))}) |
5 | 4 | eleq2d 2825 | . 2 ⊢ (𝑁 ∈ 𝑉 → (𝑋 ∈ 𝑈 ↔ 𝑋 ∈ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))})) |
6 | neeq1 3001 | . . . . . 6 ⊢ (𝑛 = 𝑋 → (𝑛 ≠ 𝑁 ↔ 𝑋 ≠ 𝑁)) | |
7 | eleq1 2827 | . . . . . 6 ⊢ (𝑛 = 𝑋 → (𝑛 ∈ 𝑒 ↔ 𝑋 ∈ 𝑒)) | |
8 | 6, 7 | 3anbi13d 1437 | . . . . 5 ⊢ (𝑛 = 𝑋 → ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ↔ (𝑋 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑋 ∈ 𝑒))) |
9 | eqeq1 2739 | . . . . . 6 ⊢ (𝑛 = 𝑋 → (𝑛 = 𝑁 ↔ 𝑋 = 𝑁)) | |
10 | sneq 4641 | . . . . . . 7 ⊢ (𝑛 = 𝑋 → {𝑛} = {𝑋}) | |
11 | 10 | eqeq2d 2746 | . . . . . 6 ⊢ (𝑛 = 𝑋 → (𝑒 = {𝑛} ↔ 𝑒 = {𝑋})) |
12 | 9, 11 | anbi12d 632 | . . . . 5 ⊢ (𝑛 = 𝑋 → ((𝑛 = 𝑁 ∧ 𝑒 = {𝑛}) ↔ (𝑋 = 𝑁 ∧ 𝑒 = {𝑋}))) |
13 | 8, 12 | orbi12d 918 | . . . 4 ⊢ (𝑛 = 𝑋 → (((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) ↔ ((𝑋 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑋 ∈ 𝑒) ∨ (𝑋 = 𝑁 ∧ 𝑒 = {𝑋})))) |
14 | 13 | rexbidv 3177 | . . 3 ⊢ (𝑛 = 𝑋 → (∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) ↔ ∃𝑒 ∈ 𝐸 ((𝑋 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑋 ∈ 𝑒) ∨ (𝑋 = 𝑁 ∧ 𝑒 = {𝑋})))) |
15 | 14 | elrab 3695 | . 2 ⊢ (𝑋 ∈ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))} ↔ (𝑋 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 ((𝑋 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑋 ∈ 𝑒) ∨ (𝑋 = 𝑁 ∧ 𝑒 = {𝑋})))) |
16 | 5, 15 | bitrdi 287 | 1 ⊢ (𝑁 ∈ 𝑉 → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 ((𝑋 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑋 ∈ 𝑒) ∨ (𝑋 = 𝑁 ∧ 𝑒 = {𝑋}))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∃wrex 3068 {crab 3433 {csn 4631 ‘cfv 6563 (class class class)co 7431 Vtxcvtx 29028 Edgcedg 29079 NeighbVtx cnbgr 29364 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-nbgr 29365 |
This theorem is referenced by: vopnbgrelself 47779 |
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