![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > nbusgredgeu0 | Structured version Visualization version GIF version |
Description: For each neighbor of a vertex there is exactly one edge between the vertex and its neighbor in a simple graph. (Contributed by Alexander van der Vekens, 17-Dec-2017.) (Revised by AV, 27-Oct-2020.) (Proof shortened by AV, 13-Feb-2022.) |
Ref | Expression |
---|---|
nbusgrf1o1.v | ⊢ 𝑉 = (Vtx‘𝐺) |
nbusgrf1o1.e | ⊢ 𝐸 = (Edg‘𝐺) |
nbusgrf1o1.n | ⊢ 𝑁 = (𝐺 NeighbVtx 𝑈) |
nbusgrf1o1.i | ⊢ 𝐼 = {𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒} |
Ref | Expression |
---|---|
nbusgredgeu0 | ⊢ (((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) ∧ 𝑀 ∈ 𝑁) → ∃!𝑖 ∈ 𝐼 𝑖 = {𝑈, 𝑀}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 765 | . . . . 5 ⊢ (((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) ∧ 𝑀 ∈ 𝑁) → 𝐺 ∈ USGraph) | |
2 | nbusgrf1o1.n | . . . . . . . 8 ⊢ 𝑁 = (𝐺 NeighbVtx 𝑈) | |
3 | 2 | eleq2i 2817 | . . . . . . 7 ⊢ (𝑀 ∈ 𝑁 ↔ 𝑀 ∈ (𝐺 NeighbVtx 𝑈)) |
4 | nbgrsym 29248 | . . . . . . . . 9 ⊢ (𝑀 ∈ (𝐺 NeighbVtx 𝑈) ↔ 𝑈 ∈ (𝐺 NeighbVtx 𝑀)) | |
5 | 4 | a1i 11 | . . . . . . . 8 ⊢ ((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) → (𝑀 ∈ (𝐺 NeighbVtx 𝑈) ↔ 𝑈 ∈ (𝐺 NeighbVtx 𝑀))) |
6 | 5 | biimpd 228 | . . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) → (𝑀 ∈ (𝐺 NeighbVtx 𝑈) → 𝑈 ∈ (𝐺 NeighbVtx 𝑀))) |
7 | 3, 6 | biimtrid 241 | . . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) → (𝑀 ∈ 𝑁 → 𝑈 ∈ (𝐺 NeighbVtx 𝑀))) |
8 | 7 | imp 405 | . . . . 5 ⊢ (((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) ∧ 𝑀 ∈ 𝑁) → 𝑈 ∈ (𝐺 NeighbVtx 𝑀)) |
9 | nbusgrf1o1.e | . . . . . 6 ⊢ 𝐸 = (Edg‘𝐺) | |
10 | 9 | nbusgredgeu 29251 | . . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑈 ∈ (𝐺 NeighbVtx 𝑀)) → ∃!𝑖 ∈ 𝐸 𝑖 = {𝑈, 𝑀}) |
11 | 1, 8, 10 | syl2anc 582 | . . . 4 ⊢ (((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) ∧ 𝑀 ∈ 𝑁) → ∃!𝑖 ∈ 𝐸 𝑖 = {𝑈, 𝑀}) |
12 | df-reu 3364 | . . . 4 ⊢ (∃!𝑖 ∈ 𝐸 𝑖 = {𝑈, 𝑀} ↔ ∃!𝑖(𝑖 ∈ 𝐸 ∧ 𝑖 = {𝑈, 𝑀})) | |
13 | 11, 12 | sylib 217 | . . 3 ⊢ (((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) ∧ 𝑀 ∈ 𝑁) → ∃!𝑖(𝑖 ∈ 𝐸 ∧ 𝑖 = {𝑈, 𝑀})) |
14 | anass 467 | . . . . 5 ⊢ (((𝑖 ∈ 𝐸 ∧ 𝑈 ∈ 𝑖) ∧ 𝑖 = {𝑈, 𝑀}) ↔ (𝑖 ∈ 𝐸 ∧ (𝑈 ∈ 𝑖 ∧ 𝑖 = {𝑈, 𝑀}))) | |
15 | prid1g 4766 | . . . . . . . . . 10 ⊢ (𝑈 ∈ 𝑉 → 𝑈 ∈ {𝑈, 𝑀}) | |
16 | 15 | ad2antlr 725 | . . . . . . . . 9 ⊢ (((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) ∧ 𝑀 ∈ 𝑁) → 𝑈 ∈ {𝑈, 𝑀}) |
17 | eleq2 2814 | . . . . . . . . 9 ⊢ (𝑖 = {𝑈, 𝑀} → (𝑈 ∈ 𝑖 ↔ 𝑈 ∈ {𝑈, 𝑀})) | |
18 | 16, 17 | syl5ibrcom 246 | . . . . . . . 8 ⊢ (((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) ∧ 𝑀 ∈ 𝑁) → (𝑖 = {𝑈, 𝑀} → 𝑈 ∈ 𝑖)) |
19 | 18 | pm4.71rd 561 | . . . . . . 7 ⊢ (((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) ∧ 𝑀 ∈ 𝑁) → (𝑖 = {𝑈, 𝑀} ↔ (𝑈 ∈ 𝑖 ∧ 𝑖 = {𝑈, 𝑀}))) |
20 | 19 | bicomd 222 | . . . . . 6 ⊢ (((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) ∧ 𝑀 ∈ 𝑁) → ((𝑈 ∈ 𝑖 ∧ 𝑖 = {𝑈, 𝑀}) ↔ 𝑖 = {𝑈, 𝑀})) |
21 | 20 | anbi2d 628 | . . . . 5 ⊢ (((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) ∧ 𝑀 ∈ 𝑁) → ((𝑖 ∈ 𝐸 ∧ (𝑈 ∈ 𝑖 ∧ 𝑖 = {𝑈, 𝑀})) ↔ (𝑖 ∈ 𝐸 ∧ 𝑖 = {𝑈, 𝑀}))) |
22 | 14, 21 | bitrid 282 | . . . 4 ⊢ (((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) ∧ 𝑀 ∈ 𝑁) → (((𝑖 ∈ 𝐸 ∧ 𝑈 ∈ 𝑖) ∧ 𝑖 = {𝑈, 𝑀}) ↔ (𝑖 ∈ 𝐸 ∧ 𝑖 = {𝑈, 𝑀}))) |
23 | 22 | eubidv 2574 | . . 3 ⊢ (((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) ∧ 𝑀 ∈ 𝑁) → (∃!𝑖((𝑖 ∈ 𝐸 ∧ 𝑈 ∈ 𝑖) ∧ 𝑖 = {𝑈, 𝑀}) ↔ ∃!𝑖(𝑖 ∈ 𝐸 ∧ 𝑖 = {𝑈, 𝑀}))) |
24 | 13, 23 | mpbird 256 | . 2 ⊢ (((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) ∧ 𝑀 ∈ 𝑁) → ∃!𝑖((𝑖 ∈ 𝐸 ∧ 𝑈 ∈ 𝑖) ∧ 𝑖 = {𝑈, 𝑀})) |
25 | df-reu 3364 | . . 3 ⊢ (∃!𝑖 ∈ 𝐼 𝑖 = {𝑈, 𝑀} ↔ ∃!𝑖(𝑖 ∈ 𝐼 ∧ 𝑖 = {𝑈, 𝑀})) | |
26 | eleq2 2814 | . . . . . 6 ⊢ (𝑒 = 𝑖 → (𝑈 ∈ 𝑒 ↔ 𝑈 ∈ 𝑖)) | |
27 | nbusgrf1o1.i | . . . . . 6 ⊢ 𝐼 = {𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒} | |
28 | 26, 27 | elrab2 3682 | . . . . 5 ⊢ (𝑖 ∈ 𝐼 ↔ (𝑖 ∈ 𝐸 ∧ 𝑈 ∈ 𝑖)) |
29 | 28 | anbi1i 622 | . . . 4 ⊢ ((𝑖 ∈ 𝐼 ∧ 𝑖 = {𝑈, 𝑀}) ↔ ((𝑖 ∈ 𝐸 ∧ 𝑈 ∈ 𝑖) ∧ 𝑖 = {𝑈, 𝑀})) |
30 | 29 | eubii 2573 | . . 3 ⊢ (∃!𝑖(𝑖 ∈ 𝐼 ∧ 𝑖 = {𝑈, 𝑀}) ↔ ∃!𝑖((𝑖 ∈ 𝐸 ∧ 𝑈 ∈ 𝑖) ∧ 𝑖 = {𝑈, 𝑀})) |
31 | 25, 30 | bitri 274 | . 2 ⊢ (∃!𝑖 ∈ 𝐼 𝑖 = {𝑈, 𝑀} ↔ ∃!𝑖((𝑖 ∈ 𝐸 ∧ 𝑈 ∈ 𝑖) ∧ 𝑖 = {𝑈, 𝑀})) |
32 | 24, 31 | sylibr 233 | 1 ⊢ (((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) ∧ 𝑀 ∈ 𝑁) → ∃!𝑖 ∈ 𝐼 𝑖 = {𝑈, 𝑀}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∃!weu 2556 ∃!wreu 3361 {crab 3418 {cpr 4632 ‘cfv 6549 (class class class)co 7419 Vtxcvtx 28881 Edgcedg 28932 USGraphcusgr 29034 NeighbVtx cnbgr 29217 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-oadd 8491 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-dju 9926 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-n0 12506 df-xnn0 12578 df-z 12592 df-uz 12856 df-fz 13520 df-hash 14326 df-edg 28933 df-upgr 28967 df-umgr 28968 df-usgr 29036 df-nbgr 29218 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |