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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 28326). (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 2737 | . . . 4 ⊢ (Edg‘𝐺) = (Edg‘𝐺) | |
3 | 1, 2 | nbgrval 28326 | . . 3 ⊢ (𝑁 ∈ 𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒}) |
4 | 3 | adantr 482 | . 2 ⊢ ((𝑁 ∈ 𝑉 ∧ Fun 𝐼) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒}) |
5 | edgval 28042 | . . . . . 6 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺) | |
6 | dfnbgr3.i | . . . . . . . 8 ⊢ 𝐼 = (iEdg‘𝐺) | |
7 | 6 | eqcomi 2746 | . . . . . . 7 ⊢ (iEdg‘𝐺) = 𝐼 |
8 | 7 | rneqi 5897 | . . . . . 6 ⊢ ran (iEdg‘𝐺) = ran 𝐼 |
9 | 5, 8 | eqtri 2765 | . . . . 5 ⊢ (Edg‘𝐺) = ran 𝐼 |
10 | 9 | rexeqi 3315 | . . . 4 ⊢ (∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ ran 𝐼{𝑁, 𝑛} ⊆ 𝑒) |
11 | funfn 6536 | . . . . . . 7 ⊢ (Fun 𝐼 ↔ 𝐼 Fn dom 𝐼) | |
12 | 11 | biimpi 215 | . . . . . 6 ⊢ (Fun 𝐼 → 𝐼 Fn dom 𝐼) |
13 | 12 | adantl 483 | . . . . 5 ⊢ ((𝑁 ∈ 𝑉 ∧ Fun 𝐼) → 𝐼 Fn dom 𝐼) |
14 | sseq2 3975 | . . . . . 6 ⊢ (𝑒 = (𝐼‘𝑖) → ({𝑁, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ⊆ (𝐼‘𝑖))) | |
15 | 14 | rexrn 7042 | . . . . 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 283 | . . 3 ⊢ ((𝑁 ∈ 𝑉 ∧ Fun 𝐼) → (∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼‘𝑖))) |
18 | 17 | rabbidv 3418 | . 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 2777 | 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 397 = wceq 1542 ∈ wcel 2107 ∃wrex 3074 {crab 3410 ∖ cdif 3912 ⊆ wss 3915 {csn 4591 {cpr 4593 dom cdm 5638 ran crn 5639 Fun wfun 6495 Fn wfn 6496 ‘cfv 6501 (class class class)co 7362 Vtxcvtx 27989 iEdgciedg 27990 Edgcedg 28040 NeighbVtx cnbgr 28322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-iota 6453 df-fun 6503 df-fn 6504 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-edg 28041 df-nbgr 28323 |
This theorem is referenced by: (None) |
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