![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 26633). (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 2825 | . . . 4 ⊢ (Edg‘𝐺) = (Edg‘𝐺) | |
3 | 1, 2 | nbgrval 26633 | . . 3 ⊢ (𝑁 ∈ 𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒}) |
4 | 3 | adantr 474 | . 2 ⊢ ((𝑁 ∈ 𝑉 ∧ Fun 𝐼) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒}) |
5 | edgval 26347 | . . . . . 6 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺) | |
6 | dfnbgr3.i | . . . . . . . 8 ⊢ 𝐼 = (iEdg‘𝐺) | |
7 | 6 | eqcomi 2834 | . . . . . . 7 ⊢ (iEdg‘𝐺) = 𝐼 |
8 | 7 | rneqi 5584 | . . . . . 6 ⊢ ran (iEdg‘𝐺) = ran 𝐼 |
9 | 5, 8 | eqtri 2849 | . . . . 5 ⊢ (Edg‘𝐺) = ran 𝐼 |
10 | 9 | rexeqi 3355 | . . . 4 ⊢ (∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ ran 𝐼{𝑁, 𝑛} ⊆ 𝑒) |
11 | funfn 6153 | . . . . . . 7 ⊢ (Fun 𝐼 ↔ 𝐼 Fn dom 𝐼) | |
12 | 11 | biimpi 208 | . . . . . 6 ⊢ (Fun 𝐼 → 𝐼 Fn dom 𝐼) |
13 | 12 | adantl 475 | . . . . 5 ⊢ ((𝑁 ∈ 𝑉 ∧ Fun 𝐼) → 𝐼 Fn dom 𝐼) |
14 | sseq2 3852 | . . . . . 6 ⊢ (𝑒 = (𝐼‘𝑖) → ({𝑁, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ⊆ (𝐼‘𝑖))) | |
15 | 14 | rexrn 6610 | . . . . 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 275 | . . 3 ⊢ ((𝑁 ∈ 𝑉 ∧ Fun 𝐼) → (∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼‘𝑖))) |
18 | 17 | rabbidv 3402 | . 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 2861 | 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 198 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ∃wrex 3118 {crab 3121 ∖ cdif 3795 ⊆ wss 3798 {csn 4397 {cpr 4399 dom cdm 5342 ran crn 5343 Fun wfun 6117 Fn wfn 6118 ‘cfv 6123 (class class class)co 6905 Vtxcvtx 26294 iEdgciedg 26295 Edgcedg 26345 NeighbVtx cnbgr 26629 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-iota 6086 df-fun 6125 df-fn 6126 df-fv 6131 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-edg 26346 df-nbgr 26630 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |