Theorem usgredg4 27584

Description: For a vertex incident to an edge there is another vertex incident to the edge. (Contributed by Alexander van der Vekens, 18-Dec-2017.) (Revised by AV, 17-Oct-2020.)

Hypotheses

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

Assertion

Ref	Expression
usgredg4	⊢ ((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 usgredg4

Dummy variables 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		usgredg3.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2		usgredg3.e	. . . 4 ⊢ 𝐸 = (iEdg‘𝐺)
3	1, 2	usgredg3 27583	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → ∃𝑥 ∈ 𝑉 ∃𝑧 ∈ 𝑉 (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧}))
4		eleq2 2827	. . . . . . . 8 ⊢ ((𝐸‘𝑋) = {𝑥, 𝑧} → (𝑌 ∈ (𝐸‘𝑋) ↔ 𝑌 ∈ {𝑥, 𝑧}))
5	4	adantl 482	. . . . . . 7 ⊢ ((𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧}) → (𝑌 ∈ (𝐸‘𝑋) ↔ 𝑌 ∈ {𝑥, 𝑧}))
6	5	adantl 482	. . . . . 6 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧})) → (𝑌 ∈ (𝐸‘𝑋) ↔ 𝑌 ∈ {𝑥, 𝑧}))
7		simplrr 775	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧})) → 𝑧 ∈ 𝑉)
8	7	adantl 482	. . . . . . . . . . 11 ⊢ ((𝑌 = 𝑥 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧}))) → 𝑧 ∈ 𝑉)
9		preq2 4670	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → {𝑥, 𝑦} = {𝑥, 𝑧})
10	9	eqeq2d 2749	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → ({𝑥, 𝑧} = {𝑥, 𝑦} ↔ {𝑥, 𝑧} = {𝑥, 𝑧}))
11	10	adantl 482	. . . . . . . . . . 11 ⊢ (((𝑌 = 𝑥 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧}))) ∧ 𝑦 = 𝑧) → ({𝑥, 𝑧} = {𝑥, 𝑦} ↔ {𝑥, 𝑧} = {𝑥, 𝑧}))
12		eqidd 2739	. . . . . . . . . . 11 ⊢ ((𝑌 = 𝑥 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧}))) → {𝑥, 𝑧} = {𝑥, 𝑧})
13	8, 11, 12	rspcedvd 3563	. . . . . . . . . 10 ⊢ ((𝑌 = 𝑥 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧}))) → ∃𝑦 ∈ 𝑉 {𝑥, 𝑧} = {𝑥, 𝑦})
14		simprr 770	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧})) → (𝐸‘𝑋) = {𝑥, 𝑧})
15		preq1 4669	. . . . . . . . . . . 12 ⊢ (𝑌 = 𝑥 → {𝑌, 𝑦} = {𝑥, 𝑦})
16	14, 15	eqeqan12rd 2753	. . . . . . . . . . 11 ⊢ ((𝑌 = 𝑥 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧}))) → ((𝐸‘𝑋) = {𝑌, 𝑦} ↔ {𝑥, 𝑧} = {𝑥, 𝑦}))
17	16	rexbidv 3226	. . . . . . . . . 10 ⊢ ((𝑌 = 𝑥 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧}))) → (∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦} ↔ ∃𝑦 ∈ 𝑉 {𝑥, 𝑧} = {𝑥, 𝑦}))
18	13, 17	mpbird 256	. . . . . . . . 9 ⊢ ((𝑌 = 𝑥 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧}))) → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦})
19	18	ex 413	. . . . . . . 8 ⊢ (𝑌 = 𝑥 → ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧})) → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦}))
20		simplrl 774	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧})) → 𝑥 ∈ 𝑉)
21	20	adantl 482	. . . . . . . . . . 11 ⊢ ((𝑌 = 𝑧 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧}))) → 𝑥 ∈ 𝑉)
22		preq2 4670	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → {𝑧, 𝑦} = {𝑧, 𝑥})
23	22	eqeq2d 2749	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → ({𝑥, 𝑧} = {𝑧, 𝑦} ↔ {𝑥, 𝑧} = {𝑧, 𝑥}))
24	23	adantl 482	. . . . . . . . . . 11 ⊢ (((𝑌 = 𝑧 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧}))) ∧ 𝑦 = 𝑥) → ({𝑥, 𝑧} = {𝑧, 𝑦} ↔ {𝑥, 𝑧} = {𝑧, 𝑥}))
25		prcom 4668	. . . . . . . . . . . 12 ⊢ {𝑥, 𝑧} = {𝑧, 𝑥}
26	25	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑌 = 𝑧 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧}))) → {𝑥, 𝑧} = {𝑧, 𝑥})
27	21, 24, 26	rspcedvd 3563	. . . . . . . . . 10 ⊢ ((𝑌 = 𝑧 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧}))) → ∃𝑦 ∈ 𝑉 {𝑥, 𝑧} = {𝑧, 𝑦})
28		preq1 4669	. . . . . . . . . . . 12 ⊢ (𝑌 = 𝑧 → {𝑌, 𝑦} = {𝑧, 𝑦})
29	14, 28	eqeqan12rd 2753	. . . . . . . . . . 11 ⊢ ((𝑌 = 𝑧 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧}))) → ((𝐸‘𝑋) = {𝑌, 𝑦} ↔ {𝑥, 𝑧} = {𝑧, 𝑦}))
30	29	rexbidv 3226	. . . . . . . . . 10 ⊢ ((𝑌 = 𝑧 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧}))) → (∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦} ↔ ∃𝑦 ∈ 𝑉 {𝑥, 𝑧} = {𝑧, 𝑦}))
31	27, 30	mpbird 256	. . . . . . . . 9 ⊢ ((𝑌 = 𝑧 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧}))) → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦})
32	31	ex 413	. . . . . . . 8 ⊢ (𝑌 = 𝑧 → ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧})) → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦}))
33	19, 32	jaoi 854	. . . . . . 7 ⊢ ((𝑌 = 𝑥 ∨ 𝑌 = 𝑧) → ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧})) → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦}))
34		elpri 4583	. . . . . . 7 ⊢ (𝑌 ∈ {𝑥, 𝑧} → (𝑌 = 𝑥 ∨ 𝑌 = 𝑧))
35	33, 34	syl11 33	. . . . . 6 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧})) → (𝑌 ∈ {𝑥, 𝑧} → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦}))
36	6, 35	sylbid 239	. . . . 5 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧})) → (𝑌 ∈ (𝐸‘𝑋) → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦}))
37	36	ex 413	. . . 4 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧}) → (𝑌 ∈ (𝐸‘𝑋) → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦})))
38	37	rexlimdvva 3223	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → (∃𝑥 ∈ 𝑉 ∃𝑧 ∈ 𝑉 (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧}) → (𝑌 ∈ (𝐸‘𝑋) → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦})))
39	3, 38	mpd 15	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → (𝑌 ∈ (𝐸‘𝑋) → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦}))
40	39	3impia 1116	1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸 ∧ 𝑌 ∈ (𝐸‘𝑋)) → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∃wrex 3065  {cpr 4563  dom cdm 5589  ‘cfv 6433  Vtxcvtx 27366  iEdgciedg 27367  USGraphcusgr 27519

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-oadd 8301  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-dju 9659  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240  df-hash 14045  df-edg 27418  df-umgr 27453  df-usgr 27521

This theorem is referenced by:  usgredgreu  27585