Theorem edgnbusgreu 26269

Description: For each edge incident to a vertex there is exactly one neighbor of the vertex also incident to this edge in a simple graph. (Contributed by AV, 28-Oct-2020.)

Hypotheses

Ref	Expression
edgnbusgreu.v	⊢ 𝑉 = (Vtx‘𝐺)
edgnbusgreu.e	⊢ 𝐸 = (Edg‘𝐺)
edgnbusgreu.n	⊢ 𝑁 = (𝐺 NeighbVtx 𝑀)

Assertion

Ref	Expression
edgnbusgreu	⊢ (((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉) ∧ (𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶)) → ∃!𝑛 ∈ 𝑁 𝐶 = {𝑀, 𝑛})

Distinct variable groups:   𝐶,𝑛   𝑛,𝐸   𝑛,𝐺   𝑛,𝑀   𝑛,𝑉

Allowed substitution hint:   𝑁(𝑛)

Proof of Theorem edgnbusgreu

Step	Hyp	Ref	Expression
1		simpl 473	. . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉) → 𝐺 ∈ USGraph )
2	1	adantr 481	. . . . 5 ⊢ (((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉) ∧ (𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶)) → 𝐺 ∈ USGraph )
3		edgnbusgreu.e	. . . . . . . 8 ⊢ 𝐸 = (Edg‘𝐺)
4	3	eleq2i 2693	. . . . . . 7 ⊢ (𝐶 ∈ 𝐸 ↔ 𝐶 ∈ (Edg‘𝐺))
5	4	biimpi 206	. . . . . 6 ⊢ (𝐶 ∈ 𝐸 → 𝐶 ∈ (Edg‘𝐺))
6	5	ad2antrl 764	. . . . 5 ⊢ (((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉) ∧ (𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶)) → 𝐶 ∈ (Edg‘𝐺))
7		simprr 796	. . . . 5 ⊢ (((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉) ∧ (𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶)) → 𝑀 ∈ 𝐶)
8		usgredg2vtxeu 26113	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝐶 ∈ (Edg‘𝐺) ∧ 𝑀 ∈ 𝐶) → ∃!𝑛 ∈ (Vtx‘𝐺)𝐶 = {𝑀, 𝑛})
9	2, 6, 7, 8	syl3anc 1326	. . . 4 ⊢ (((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉) ∧ (𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶)) → ∃!𝑛 ∈ (Vtx‘𝐺)𝐶 = {𝑀, 𝑛})
10		df-reu 2919	. . . . 5 ⊢ (∃!𝑛 ∈ (Vtx‘𝐺)𝐶 = {𝑀, 𝑛} ↔ ∃!𝑛(𝑛 ∈ (Vtx‘𝐺) ∧ 𝐶 = {𝑀, 𝑛}))
11		prcom 4267	. . . . . . . . . . . . . . . 16 ⊢ {𝑀, 𝑛} = {𝑛, 𝑀}
12	11	eqeq2i 2634	. . . . . . . . . . . . . . 15 ⊢ (𝐶 = {𝑀, 𝑛} ↔ 𝐶 = {𝑛, 𝑀})
13	12	biimpi 206	. . . . . . . . . . . . . 14 ⊢ (𝐶 = {𝑀, 𝑛} → 𝐶 = {𝑛, 𝑀})
14	13	eleq1d 2686	. . . . . . . . . . . . 13 ⊢ (𝐶 = {𝑀, 𝑛} → (𝐶 ∈ 𝐸 ↔ {𝑛, 𝑀} ∈ 𝐸))
15	14	biimpcd 239	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ 𝐸 → (𝐶 = {𝑀, 𝑛} → {𝑛, 𝑀} ∈ 𝐸))
16	15	ad2antrl 764	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉) ∧ (𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶)) → (𝐶 = {𝑀, 𝑛} → {𝑛, 𝑀} ∈ 𝐸))
17	16	adantld 483	. . . . . . . . . 10 ⊢ (((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉) ∧ (𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶)) → ((𝑛 ∈ (Vtx‘𝐺) ∧ 𝐶 = {𝑀, 𝑛}) → {𝑛, 𝑀} ∈ 𝐸))
18	17	imp 445	. . . . . . . . 9 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉) ∧ (𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶)) ∧ (𝑛 ∈ (Vtx‘𝐺) ∧ 𝐶 = {𝑀, 𝑛})) → {𝑛, 𝑀} ∈ 𝐸)
19		simprr 796	. . . . . . . . 9 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉) ∧ (𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶)) ∧ (𝑛 ∈ (Vtx‘𝐺) ∧ 𝐶 = {𝑀, 𝑛})) → 𝐶 = {𝑀, 𝑛})
20	18, 19	jca 554	. . . . . . . 8 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉) ∧ (𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶)) ∧ (𝑛 ∈ (Vtx‘𝐺) ∧ 𝐶 = {𝑀, 𝑛})) → ({𝑛, 𝑀} ∈ 𝐸 ∧ 𝐶 = {𝑀, 𝑛}))
21		simpl 473	. . . . . . . . . 10 ⊢ (({𝑛, 𝑀} ∈ 𝐸 ∧ 𝐶 = {𝑀, 𝑛}) → {𝑛, 𝑀} ∈ 𝐸)
22		eqid 2622	. . . . . . . . . . . 12 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
23	3, 22	usgrpredgv 26089	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ USGraph ∧ {𝑛, 𝑀} ∈ 𝐸) → (𝑛 ∈ (Vtx‘𝐺) ∧ 𝑀 ∈ (Vtx‘𝐺)))
24	23	simpld 475	. . . . . . . . . 10 ⊢ ((𝐺 ∈ USGraph ∧ {𝑛, 𝑀} ∈ 𝐸) → 𝑛 ∈ (Vtx‘𝐺))
25	2, 21, 24	syl2an 494	. . . . . . . . 9 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉) ∧ (𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶)) ∧ ({𝑛, 𝑀} ∈ 𝐸 ∧ 𝐶 = {𝑀, 𝑛})) → 𝑛 ∈ (Vtx‘𝐺))
26		simprr 796	. . . . . . . . 9 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉) ∧ (𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶)) ∧ ({𝑛, 𝑀} ∈ 𝐸 ∧ 𝐶 = {𝑀, 𝑛})) → 𝐶 = {𝑀, 𝑛})
27	25, 26	jca 554	. . . . . . . 8 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉) ∧ (𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶)) ∧ ({𝑛, 𝑀} ∈ 𝐸 ∧ 𝐶 = {𝑀, 𝑛})) → (𝑛 ∈ (Vtx‘𝐺) ∧ 𝐶 = {𝑀, 𝑛}))
28	20, 27	impbida 877	. . . . . . 7 ⊢ (((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉) ∧ (𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶)) → ((𝑛 ∈ (Vtx‘𝐺) ∧ 𝐶 = {𝑀, 𝑛}) ↔ ({𝑛, 𝑀} ∈ 𝐸 ∧ 𝐶 = {𝑀, 𝑛})))
29	28	eubidv 2490	. . . . . 6 ⊢ (((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉) ∧ (𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶)) → (∃!𝑛(𝑛 ∈ (Vtx‘𝐺) ∧ 𝐶 = {𝑀, 𝑛}) ↔ ∃!𝑛({𝑛, 𝑀} ∈ 𝐸 ∧ 𝐶 = {𝑀, 𝑛})))
30	29	biimpd 219	. . . . 5 ⊢ (((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉) ∧ (𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶)) → (∃!𝑛(𝑛 ∈ (Vtx‘𝐺) ∧ 𝐶 = {𝑀, 𝑛}) → ∃!𝑛({𝑛, 𝑀} ∈ 𝐸 ∧ 𝐶 = {𝑀, 𝑛})))
31	10, 30	syl5bi 232	. . . 4 ⊢ (((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉) ∧ (𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶)) → (∃!𝑛 ∈ (Vtx‘𝐺)𝐶 = {𝑀, 𝑛} → ∃!𝑛({𝑛, 𝑀} ∈ 𝐸 ∧ 𝐶 = {𝑀, 𝑛})))
32	9, 31	mpd 15	. . 3 ⊢ (((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉) ∧ (𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶)) → ∃!𝑛({𝑛, 𝑀} ∈ 𝐸 ∧ 𝐶 = {𝑀, 𝑛}))
33		edgnbusgreu.n	. . . . . . . 8 ⊢ 𝑁 = (𝐺 NeighbVtx 𝑀)
34	33	eleq2i 2693	. . . . . . 7 ⊢ (𝑛 ∈ 𝑁 ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝑀))
35	3	nbusgreledg 26249	. . . . . . 7 ⊢ (𝐺 ∈ USGraph → (𝑛 ∈ (𝐺 NeighbVtx 𝑀) ↔ {𝑛, 𝑀} ∈ 𝐸))
36	34, 35	syl5bb 272	. . . . . 6 ⊢ (𝐺 ∈ USGraph → (𝑛 ∈ 𝑁 ↔ {𝑛, 𝑀} ∈ 𝐸))
37	36	anbi1d 741	. . . . 5 ⊢ (𝐺 ∈ USGraph → ((𝑛 ∈ 𝑁 ∧ 𝐶 = {𝑀, 𝑛}) ↔ ({𝑛, 𝑀} ∈ 𝐸 ∧ 𝐶 = {𝑀, 𝑛})))
38	37	ad2antrr 762	. . . 4 ⊢ (((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉) ∧ (𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶)) → ((𝑛 ∈ 𝑁 ∧ 𝐶 = {𝑀, 𝑛}) ↔ ({𝑛, 𝑀} ∈ 𝐸 ∧ 𝐶 = {𝑀, 𝑛})))
39	38	eubidv 2490	. . 3 ⊢ (((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉) ∧ (𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶)) → (∃!𝑛(𝑛 ∈ 𝑁 ∧ 𝐶 = {𝑀, 𝑛}) ↔ ∃!𝑛({𝑛, 𝑀} ∈ 𝐸 ∧ 𝐶 = {𝑀, 𝑛})))
40	32, 39	mpbird 247	. 2 ⊢ (((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉) ∧ (𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶)) → ∃!𝑛(𝑛 ∈ 𝑁 ∧ 𝐶 = {𝑀, 𝑛}))
41		df-reu 2919	. 2 ⊢ (∃!𝑛 ∈ 𝑁 𝐶 = {𝑀, 𝑛} ↔ ∃!𝑛(𝑛 ∈ 𝑁 ∧ 𝐶 = {𝑀, 𝑛}))
42	40, 41	sylibr 224	1 ⊢ (((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉) ∧ (𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶)) → ∃!𝑛 ∈ 𝑁 𝐶 = {𝑀, 𝑛})

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∃!weu 2470  ∃!wreu 2914  {cpr 4179  ‘cfv 5888  (class class class)co 6650  Vtxcvtx 25874  Edgcedg 25939   USGraph cusgr 26044   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-uspgr 26045  df-usgr 26046  df-nbgr 26228

This theorem is referenced by:  nbusgrf1o0  26271