Theorem nbusgredgeu0 29339

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.)

Hypotheses

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}

Assertion

Ref	Expression
nbusgredgeu0	⊢ (((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) ∧ 𝑀 ∈ 𝑁) → ∃!𝑖 ∈ 𝐼 𝑖 = {𝑈, 𝑀})

Distinct variable groups:   𝑖,𝐸,𝑒   𝑖,𝐺   𝑖,𝑀   𝑖,𝑁   𝑈,𝑖,𝑒   𝑖,𝑉

Allowed substitution hints:   𝐺(𝑒)   𝐼(𝑒,𝑖)   𝑀(𝑒)   𝑁(𝑒)   𝑉(𝑒)

Proof of Theorem nbusgredgeu0

Step	Hyp	Ref	Expression
1		simpll 766	. . . . 5 ⊢ (((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) ∧ 𝑀 ∈ 𝑁) → 𝐺 ∈ USGraph)
2		nbusgrf1o1.n	. . . . . . . 8 ⊢ 𝑁 = (𝐺 NeighbVtx 𝑈)
3	2	eleq2i 2821	. . . . . . 7 ⊢ (𝑀 ∈ 𝑁 ↔ 𝑀 ∈ (𝐺 NeighbVtx 𝑈))
4		nbgrsym 29334	. . . . . . . . 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 229	. . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) → (𝑀 ∈ (𝐺 NeighbVtx 𝑈) → 𝑈 ∈ (𝐺 NeighbVtx 𝑀)))
7	3, 6	biimtrid 242	. . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) → (𝑀 ∈ 𝑁 → 𝑈 ∈ (𝐺 NeighbVtx 𝑀)))
8	7	imp 406	. . . . 5 ⊢ (((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) ∧ 𝑀 ∈ 𝑁) → 𝑈 ∈ (𝐺 NeighbVtx 𝑀))
9		nbusgrf1o1.e	. . . . . 6 ⊢ 𝐸 = (Edg‘𝐺)
10	9	nbusgredgeu 29337	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑈 ∈ (𝐺 NeighbVtx 𝑀)) → ∃!𝑖 ∈ 𝐸 𝑖 = {𝑈, 𝑀})
11	1, 8, 10	syl2anc 584	. . . 4 ⊢ (((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) ∧ 𝑀 ∈ 𝑁) → ∃!𝑖 ∈ 𝐸 𝑖 = {𝑈, 𝑀})
12		df-reu 3345	. . . 4 ⊢ (∃!𝑖 ∈ 𝐸 𝑖 = {𝑈, 𝑀} ↔ ∃!𝑖(𝑖 ∈ 𝐸 ∧ 𝑖 = {𝑈, 𝑀}))
13	11, 12	sylib 218	. . 3 ⊢ (((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) ∧ 𝑀 ∈ 𝑁) → ∃!𝑖(𝑖 ∈ 𝐸 ∧ 𝑖 = {𝑈, 𝑀}))
14		anass 468	. . . . 5 ⊢ (((𝑖 ∈ 𝐸 ∧ 𝑈 ∈ 𝑖) ∧ 𝑖 = {𝑈, 𝑀}) ↔ (𝑖 ∈ 𝐸 ∧ (𝑈 ∈ 𝑖 ∧ 𝑖 = {𝑈, 𝑀})))
15		prid1g 4711	. . . . . . . . . 10 ⊢ (𝑈 ∈ 𝑉 → 𝑈 ∈ {𝑈, 𝑀})
16	15	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) ∧ 𝑀 ∈ 𝑁) → 𝑈 ∈ {𝑈, 𝑀})
17		eleq2 2818	. . . . . . . . 9 ⊢ (𝑖 = {𝑈, 𝑀} → (𝑈 ∈ 𝑖 ↔ 𝑈 ∈ {𝑈, 𝑀}))
18	16, 17	syl5ibrcom 247	. . . . . . . 8 ⊢ (((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) ∧ 𝑀 ∈ 𝑁) → (𝑖 = {𝑈, 𝑀} → 𝑈 ∈ 𝑖))
19	18	pm4.71rd 562	. . . . . . 7 ⊢ (((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) ∧ 𝑀 ∈ 𝑁) → (𝑖 = {𝑈, 𝑀} ↔ (𝑈 ∈ 𝑖 ∧ 𝑖 = {𝑈, 𝑀})))
20	19	bicomd 223	. . . . . 6 ⊢ (((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) ∧ 𝑀 ∈ 𝑁) → ((𝑈 ∈ 𝑖 ∧ 𝑖 = {𝑈, 𝑀}) ↔ 𝑖 = {𝑈, 𝑀}))
21	20	anbi2d 630	. . . . 5 ⊢ (((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) ∧ 𝑀 ∈ 𝑁) → ((𝑖 ∈ 𝐸 ∧ (𝑈 ∈ 𝑖 ∧ 𝑖 = {𝑈, 𝑀})) ↔ (𝑖 ∈ 𝐸 ∧ 𝑖 = {𝑈, 𝑀})))
22	14, 21	bitrid 283	. . . 4 ⊢ (((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) ∧ 𝑀 ∈ 𝑁) → (((𝑖 ∈ 𝐸 ∧ 𝑈 ∈ 𝑖) ∧ 𝑖 = {𝑈, 𝑀}) ↔ (𝑖 ∈ 𝐸 ∧ 𝑖 = {𝑈, 𝑀})))
23	22	eubidv 2580	. . 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 257	. 2 ⊢ (((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) ∧ 𝑀 ∈ 𝑁) → ∃!𝑖((𝑖 ∈ 𝐸 ∧ 𝑈 ∈ 𝑖) ∧ 𝑖 = {𝑈, 𝑀}))
25		df-reu 3345	. . 3 ⊢ (∃!𝑖 ∈ 𝐼 𝑖 = {𝑈, 𝑀} ↔ ∃!𝑖(𝑖 ∈ 𝐼 ∧ 𝑖 = {𝑈, 𝑀}))
26		eleq2 2818	. . . . . 6 ⊢ (𝑒 = 𝑖 → (𝑈 ∈ 𝑒 ↔ 𝑈 ∈ 𝑖))
27		nbusgrf1o1.i	. . . . . 6 ⊢ 𝐼 = {𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒}
28	26, 27	elrab2 3648	. . . . 5 ⊢ (𝑖 ∈ 𝐼 ↔ (𝑖 ∈ 𝐸 ∧ 𝑈 ∈ 𝑖))
29	28	anbi1i 624	. . . 4 ⊢ ((𝑖 ∈ 𝐼 ∧ 𝑖 = {𝑈, 𝑀}) ↔ ((𝑖 ∈ 𝐸 ∧ 𝑈 ∈ 𝑖) ∧ 𝑖 = {𝑈, 𝑀}))
30	29	eubii 2579	. . 3 ⊢ (∃!𝑖(𝑖 ∈ 𝐼 ∧ 𝑖 = {𝑈, 𝑀}) ↔ ∃!𝑖((𝑖 ∈ 𝐸 ∧ 𝑈 ∈ 𝑖) ∧ 𝑖 = {𝑈, 𝑀}))
31	25, 30	bitri 275	. 2 ⊢ (∃!𝑖 ∈ 𝐼 𝑖 = {𝑈, 𝑀} ↔ ∃!𝑖((𝑖 ∈ 𝐸 ∧ 𝑈 ∈ 𝑖) ∧ 𝑖 = {𝑈, 𝑀}))
32	24, 31	sylibr 234	1 ⊢ (((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) ∧ 𝑀 ∈ 𝑁) → ∃!𝑖 ∈ 𝐼 𝑖 = {𝑈, 𝑀})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2110  ∃!weu 2562  ∃!wreu 3342  {crab 3393  {cpr 4576  ‘cfv 6477  (class class class)co 7341  Vtxcvtx 28967  Edgcedg 29018  USGraphcusgr 29120   NeighbVtx cnbgr 29303

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663  ax-cnex 11054  ax-resscn 11055  ax-1cn 11056  ax-icn 11057  ax-addcl 11058  ax-addrcl 11059  ax-mulcl 11060  ax-mulrcl 11061  ax-mulcom 11062  ax-addass 11063  ax-mulass 11064  ax-distr 11065  ax-i2m1 11066  ax-1ne0 11067  ax-1rid 11068  ax-rnegex 11069  ax-rrecex 11070  ax-cnre 11071  ax-pre-lttri 11072  ax-pre-lttrn 11073  ax-pre-ltadd 11074  ax-pre-mulgt0 11075

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3344  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6244  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-1st 7916  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-2o 8381  df-oadd 8384  df-er 8617  df-en 8865  df-dom 8866  df-sdom 8867  df-fin 8868  df-dju 9786  df-card 9824  df-pnf 11140  df-mnf 11141  df-xr 11142  df-ltxr 11143  df-le 11144  df-sub 11338  df-neg 11339  df-nn 12118  df-2 12180  df-n0 12374  df-xnn0 12447  df-z 12461  df-uz 12725  df-fz 13400  df-hash 14230  df-edg 29019  df-upgr 29053  df-umgr 29054  df-usgr 29122  df-nbgr 29304

This theorem is referenced by: (None)