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Mirrors > Home > MPE Home > Th. List > dfnbgr3 | Structured version Visualization version GIF version |
Description: Alternate definition of the neighbors of a vertex using the edge function instead of the edges themselves (see also nbgrval 27120). (Contributed by Alexander van der Vekens, 17-Dec-2017.) (Revised by AV, 25-Oct-2020.) (Revised by AV, 21-Mar-2021.) |
Ref | Expression |
---|---|
dfnbgr3.v | ⊢ 𝑉 = (Vtx‘𝐺) |
dfnbgr3.i | ⊢ 𝐼 = (iEdg‘𝐺) |
Ref | Expression |
---|---|
dfnbgr3 | ⊢ ((𝑁 ∈ 𝑉 ∧ Fun 𝐼) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼‘𝑖)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfnbgr3.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | eqid 2823 | . . . 4 ⊢ (Edg‘𝐺) = (Edg‘𝐺) | |
3 | 1, 2 | nbgrval 27120 | . . 3 ⊢ (𝑁 ∈ 𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒}) |
4 | 3 | adantr 483 | . 2 ⊢ ((𝑁 ∈ 𝑉 ∧ Fun 𝐼) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒}) |
5 | edgval 26836 | . . . . . 6 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺) | |
6 | dfnbgr3.i | . . . . . . . 8 ⊢ 𝐼 = (iEdg‘𝐺) | |
7 | 6 | eqcomi 2832 | . . . . . . 7 ⊢ (iEdg‘𝐺) = 𝐼 |
8 | 7 | rneqi 5809 | . . . . . 6 ⊢ ran (iEdg‘𝐺) = ran 𝐼 |
9 | 5, 8 | eqtri 2846 | . . . . 5 ⊢ (Edg‘𝐺) = ran 𝐼 |
10 | 9 | rexeqi 3416 | . . . 4 ⊢ (∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ ran 𝐼{𝑁, 𝑛} ⊆ 𝑒) |
11 | funfn 6387 | . . . . . . 7 ⊢ (Fun 𝐼 ↔ 𝐼 Fn dom 𝐼) | |
12 | 11 | biimpi 218 | . . . . . 6 ⊢ (Fun 𝐼 → 𝐼 Fn dom 𝐼) |
13 | 12 | adantl 484 | . . . . 5 ⊢ ((𝑁 ∈ 𝑉 ∧ Fun 𝐼) → 𝐼 Fn dom 𝐼) |
14 | sseq2 3995 | . . . . . 6 ⊢ (𝑒 = (𝐼‘𝑖) → ({𝑁, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ⊆ (𝐼‘𝑖))) | |
15 | 14 | rexrn 6855 | . . . . 5 ⊢ (𝐼 Fn dom 𝐼 → (∃𝑒 ∈ ran 𝐼{𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼‘𝑖))) |
16 | 13, 15 | syl 17 | . . . 4 ⊢ ((𝑁 ∈ 𝑉 ∧ Fun 𝐼) → (∃𝑒 ∈ ran 𝐼{𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼‘𝑖))) |
17 | 10, 16 | syl5bb 285 | . . 3 ⊢ ((𝑁 ∈ 𝑉 ∧ Fun 𝐼) → (∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼‘𝑖))) |
18 | 17 | rabbidv 3482 | . 2 ⊢ ((𝑁 ∈ 𝑉 ∧ Fun 𝐼) → {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒} = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼‘𝑖)}) |
19 | 4, 18 | eqtrd 2858 | 1 ⊢ ((𝑁 ∈ 𝑉 ∧ Fun 𝐼) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼‘𝑖)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3141 {crab 3144 ∖ cdif 3935 ⊆ wss 3938 {csn 4569 {cpr 4571 dom cdm 5557 ran crn 5558 Fun wfun 6351 Fn wfn 6352 ‘cfv 6357 (class class class)co 7158 Vtxcvtx 26783 iEdgciedg 26784 Edgcedg 26834 NeighbVtx cnbgr 27116 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-iota 6316 df-fun 6359 df-fn 6360 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-edg 26835 df-nbgr 27117 |
This theorem is referenced by: (None) |
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