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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 29267). (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 2726 | . . . 4 ⊢ (Edg‘𝐺) = (Edg‘𝐺) | |
3 | 1, 2 | nbgrval 29267 | . . 3 ⊢ (𝑁 ∈ 𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒}) |
4 | 3 | adantr 479 | . 2 ⊢ ((𝑁 ∈ 𝑉 ∧ Fun 𝐼) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒}) |
5 | edgval 28980 | . . . . . 6 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺) | |
6 | dfnbgr3.i | . . . . . . . 8 ⊢ 𝐼 = (iEdg‘𝐺) | |
7 | 6 | eqcomi 2735 | . . . . . . 7 ⊢ (iEdg‘𝐺) = 𝐼 |
8 | 7 | rneqi 5934 | . . . . . 6 ⊢ ran (iEdg‘𝐺) = ran 𝐼 |
9 | 5, 8 | eqtri 2754 | . . . . 5 ⊢ (Edg‘𝐺) = ran 𝐼 |
10 | 9 | rexeqi 3314 | . . . 4 ⊢ (∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ ran 𝐼{𝑁, 𝑛} ⊆ 𝑒) |
11 | funfn 6579 | . . . . . . 7 ⊢ (Fun 𝐼 ↔ 𝐼 Fn dom 𝐼) | |
12 | 11 | biimpi 215 | . . . . . 6 ⊢ (Fun 𝐼 → 𝐼 Fn dom 𝐼) |
13 | 12 | adantl 480 | . . . . 5 ⊢ ((𝑁 ∈ 𝑉 ∧ Fun 𝐼) → 𝐼 Fn dom 𝐼) |
14 | sseq2 4006 | . . . . . 6 ⊢ (𝑒 = (𝐼‘𝑖) → ({𝑁, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ⊆ (𝐼‘𝑖))) | |
15 | 14 | rexrn 7091 | . . . . 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 | bitrid 282 | . . 3 ⊢ ((𝑁 ∈ 𝑉 ∧ Fun 𝐼) → (∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼‘𝑖))) |
18 | 17 | rabbidv 3428 | . 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 2766 | 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 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∃wrex 3060 {crab 3420 ∖ cdif 3944 ⊆ wss 3947 {csn 4624 {cpr 4626 dom cdm 5673 ran crn 5674 Fun wfun 6538 Fn wfn 6539 ‘cfv 6544 (class class class)co 7414 Vtxcvtx 28927 iEdgciedg 28928 Edgcedg 28978 NeighbVtx cnbgr 29263 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5295 ax-nul 5302 ax-pr 5424 ax-un 7736 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3421 df-v 3465 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-iota 6496 df-fun 6546 df-fn 6547 df-fv 6552 df-ov 7417 df-oprab 7418 df-mpo 7419 df-edg 28979 df-nbgr 29264 |
This theorem is referenced by: (None) |
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