Theorem usgredgreu 15979

Description: For a vertex incident to an edge there is exactly one other vertex incident to the edge. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.)

Hypotheses

Ref	Expression
usgredg3.v	⊢ 𝑉 = (Vtx‘𝐺)
usgredg3.e	⊢ 𝐸 = (iEdg‘𝐺)

Assertion

Ref	Expression
usgredgreu	⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸 ∧ 𝑌 ∈ (𝐸‘𝑋)) → ∃!𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦})

Distinct variable groups:   𝑦,𝐸   𝑦,𝐺   𝑦,𝑉   𝑦,𝑋   𝑦,𝑌

Proof of Theorem usgredgreu

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		usgredg3.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2		usgredg3.e	. . 3 ⊢ 𝐸 = (iEdg‘𝐺)
3	1, 2	usgredg4 15978	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸 ∧ 𝑌 ∈ (𝐸‘𝑋)) → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦})
4		eqtr2 2228	. . . . 5 ⊢ (((𝐸‘𝑋) = {𝑌, 𝑦} ∧ (𝐸‘𝑋) = {𝑌, 𝑥}) → {𝑌, 𝑦} = {𝑌, 𝑥})
5		vex 2782	. . . . . 6 ⊢ 𝑦 ∈ V
6		vex 2782	. . . . . 6 ⊢ 𝑥 ∈ V
7	5, 6	preqr2 3826	. . . . 5 ⊢ ({𝑌, 𝑦} = {𝑌, 𝑥} → 𝑦 = 𝑥)
8	4, 7	syl 14	. . . 4 ⊢ (((𝐸‘𝑋) = {𝑌, 𝑦} ∧ (𝐸‘𝑋) = {𝑌, 𝑥}) → 𝑦 = 𝑥)
9	8	a1i 9	. . 3 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸 ∧ 𝑌 ∈ (𝐸‘𝑋)) ∧ (𝑦 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉)) → (((𝐸‘𝑋) = {𝑌, 𝑦} ∧ (𝐸‘𝑋) = {𝑌, 𝑥}) → 𝑦 = 𝑥))
10	9	ralrimivva 2592	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸 ∧ 𝑌 ∈ (𝐸‘𝑋)) → ∀𝑦 ∈ 𝑉 ∀𝑥 ∈ 𝑉 (((𝐸‘𝑋) = {𝑌, 𝑦} ∧ (𝐸‘𝑋) = {𝑌, 𝑥}) → 𝑦 = 𝑥))
11		preq2 3724	. . . 4 ⊢ (𝑦 = 𝑥 → {𝑌, 𝑦} = {𝑌, 𝑥})
12	11	eqeq2d 2221	. . 3 ⊢ (𝑦 = 𝑥 → ((𝐸‘𝑋) = {𝑌, 𝑦} ↔ (𝐸‘𝑋) = {𝑌, 𝑥}))
13	12	reu4 2977	. 2 ⊢ (∃!𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦} ↔ (∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦} ∧ ∀𝑦 ∈ 𝑉 ∀𝑥 ∈ 𝑉 (((𝐸‘𝑋) = {𝑌, 𝑦} ∧ (𝐸‘𝑋) = {𝑌, 𝑥}) → 𝑦 = 𝑥)))
14	3, 10, 13	sylanbrc 417	1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸 ∧ 𝑌 ∈ (𝐸‘𝑋)) → ∃!𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦})

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 983   = wceq 1375   ∈ wcel 2180  ∀wral 2488  ∃wrex 2489  ∃!wreu 2490  {cpr 3647  dom cdm 4696  ‘cfv 5294  Vtxcvtx 15778  iEdgciedg 15779  USGraphcusgr 15917

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-nul 4189  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-iinf 4657  ax-cnex 8058  ax-resscn 8059  ax-1cn 8060  ax-1re 8061  ax-icn 8062  ax-addcl 8063  ax-addrcl 8064  ax-mulcl 8065  ax-addcom 8067  ax-mulcom 8068  ax-addass 8069  ax-mulass 8070  ax-distr 8071  ax-i2m1 8072  ax-1rid 8074  ax-0id 8075  ax-rnegex 8076  ax-cnre 8078

This theorem depends on definitions:  df-bi 117  df-dc 839  df-3or 984  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-ral 2493  df-rex 2494  df-reu 2495  df-rmo 2496  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-if 3583  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-br 4063  df-opab 4125  df-mpt 4126  df-tr 4162  df-id 4361  df-iord 4434  df-on 4436  df-suc 4439  df-iom 4660  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-riota 5927  df-ov 5977  df-oprab 5978  df-mpo 5979  df-1st 6256  df-2nd 6257  df-1o 6532  df-2o 6533  df-er 6650  df-en 6858  df-sub 8287  df-inn 9079  df-2 9137  df-3 9138  df-4 9139  df-5 9140  df-6 9141  df-7 9142  df-8 9143  df-9 9144  df-n0 9338  df-dec 9547  df-ndx 13001  df-slot 13002  df-base 13004  df-edgf 15771  df-vtx 15780  df-iedg 15781  df-edg 15824  df-umgren 15859  df-usgren 15919

This theorem is referenced by:  usgredg2vlem1  15985  usgredg2vlem2  15986