Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > nbumgr | Structured version Visualization version GIF version |
Description: The set of neighbors of an arbitrary class in a multigraph. (Contributed by AV, 27-Nov-2020.) |
Ref | Expression |
---|---|
nbgrel.v | ⊢ 𝑉 = (Vtx‘𝐺) |
nbgrel.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
nbumgr | ⊢ (𝐺 ∈ UMGraph → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nbgrel.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | nbgrel.e | . . . 4 ⊢ 𝐸 = (Edg‘𝐺) | |
3 | 1, 2 | nbumgrvtx 26242 | . . 3 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ 𝑉) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸}) |
4 | 3 | expcom 451 | . 2 ⊢ (𝑁 ∈ 𝑉 → (𝐺 ∈ UMGraph → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸})) |
5 | df-nel 2898 | . . . . . 6 ⊢ (𝑁 ∉ 𝑉 ↔ ¬ 𝑁 ∈ 𝑉) | |
6 | 1 | nbgrnvtx0 26237 | . . . . . 6 ⊢ (𝑁 ∉ 𝑉 → (𝐺 NeighbVtx 𝑁) = ∅) |
7 | 5, 6 | sylbir 225 | . . . . 5 ⊢ (¬ 𝑁 ∈ 𝑉 → (𝐺 NeighbVtx 𝑁) = ∅) |
8 | 7 | adantr 481 | . . . 4 ⊢ ((¬ 𝑁 ∈ 𝑉 ∧ 𝐺 ∈ UMGraph ) → (𝐺 NeighbVtx 𝑁) = ∅) |
9 | 1, 2 | umgrpredgv 26035 | . . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ UMGraph ∧ {𝑁, 𝑛} ∈ 𝐸) → (𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) |
10 | 9 | simpld 475 | . . . . . . . . . . . 12 ⊢ ((𝐺 ∈ UMGraph ∧ {𝑁, 𝑛} ∈ 𝐸) → 𝑁 ∈ 𝑉) |
11 | 10 | ex 450 | . . . . . . . . . . 11 ⊢ (𝐺 ∈ UMGraph → ({𝑁, 𝑛} ∈ 𝐸 → 𝑁 ∈ 𝑉)) |
12 | 11 | adantl 482 | . . . . . . . . . 10 ⊢ ((𝑛 ∈ 𝑉 ∧ 𝐺 ∈ UMGraph ) → ({𝑁, 𝑛} ∈ 𝐸 → 𝑁 ∈ 𝑉)) |
13 | 12 | con3d 148 | . . . . . . . . 9 ⊢ ((𝑛 ∈ 𝑉 ∧ 𝐺 ∈ UMGraph ) → (¬ 𝑁 ∈ 𝑉 → ¬ {𝑁, 𝑛} ∈ 𝐸)) |
14 | 13 | ex 450 | . . . . . . . 8 ⊢ (𝑛 ∈ 𝑉 → (𝐺 ∈ UMGraph → (¬ 𝑁 ∈ 𝑉 → ¬ {𝑁, 𝑛} ∈ 𝐸))) |
15 | 14 | com13 88 | . . . . . . 7 ⊢ (¬ 𝑁 ∈ 𝑉 → (𝐺 ∈ UMGraph → (𝑛 ∈ 𝑉 → ¬ {𝑁, 𝑛} ∈ 𝐸))) |
16 | 15 | imp 445 | . . . . . 6 ⊢ ((¬ 𝑁 ∈ 𝑉 ∧ 𝐺 ∈ UMGraph ) → (𝑛 ∈ 𝑉 → ¬ {𝑁, 𝑛} ∈ 𝐸)) |
17 | 16 | ralrimiv 2965 | . . . . 5 ⊢ ((¬ 𝑁 ∈ 𝑉 ∧ 𝐺 ∈ UMGraph ) → ∀𝑛 ∈ 𝑉 ¬ {𝑁, 𝑛} ∈ 𝐸) |
18 | rabeq0 3957 | . . . . 5 ⊢ ({𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸} = ∅ ↔ ∀𝑛 ∈ 𝑉 ¬ {𝑁, 𝑛} ∈ 𝐸) | |
19 | 17, 18 | sylibr 224 | . . . 4 ⊢ ((¬ 𝑁 ∈ 𝑉 ∧ 𝐺 ∈ UMGraph ) → {𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸} = ∅) |
20 | 8, 19 | eqtr4d 2659 | . . 3 ⊢ ((¬ 𝑁 ∈ 𝑉 ∧ 𝐺 ∈ UMGraph ) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸}) |
21 | 20 | ex 450 | . 2 ⊢ (¬ 𝑁 ∈ 𝑉 → (𝐺 ∈ UMGraph → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸})) |
22 | 4, 21 | pm2.61i 176 | 1 ⊢ (𝐺 ∈ UMGraph → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∉ wnel 2897 ∀wral 2912 {crab 2916 ∅c0 3915 {cpr 4179 ‘cfv 5888 (class class class)co 6650 Vtxcvtx 25874 Edgcedg 25939 UMGraph cumgr 25976 NeighbVtx cnbgr 26224 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-fz 12327 df-hash 13118 df-edg 25940 df-upgr 25977 df-umgr 25978 df-nbgr 26228 |
This theorem is referenced by: nbusgr 26245 |
Copyright terms: Public domain | W3C validator |