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Theorem usgredg4 27111
 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 𝐸𝑌 ∈ (𝐸𝑋)) → ∃𝑦𝑉 (𝐸𝑋) = {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.
StepHypRef Expression
1 usgredg3.v . . . 4 𝑉 = (Vtx‘𝐺)
2 usgredg3.e . . . 4 𝐸 = (iEdg‘𝐺)
31, 2usgredg3 27110 . . 3 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → ∃𝑥𝑉𝑧𝑉 (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))
4 eleq2 2840 . . . . . . . 8 ((𝐸𝑋) = {𝑥, 𝑧} → (𝑌 ∈ (𝐸𝑋) ↔ 𝑌 ∈ {𝑥, 𝑧}))
54adantl 485 . . . . . . 7 ((𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}) → (𝑌 ∈ (𝐸𝑋) ↔ 𝑌 ∈ {𝑥, 𝑧}))
65adantl 485 . . . . . 6 ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧})) → (𝑌 ∈ (𝐸𝑋) ↔ 𝑌 ∈ {𝑥, 𝑧}))
7 simplrr 777 . . . . . . . . . . . 12 ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧})) → 𝑧𝑉)
87adantl 485 . . . . . . . . . . 11 ((𝑌 = 𝑥 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → 𝑧𝑉)
9 preq2 4630 . . . . . . . . . . . . 13 (𝑦 = 𝑧 → {𝑥, 𝑦} = {𝑥, 𝑧})
109eqeq2d 2769 . . . . . . . . . . . 12 (𝑦 = 𝑧 → ({𝑥, 𝑧} = {𝑥, 𝑦} ↔ {𝑥, 𝑧} = {𝑥, 𝑧}))
1110adantl 485 . . . . . . . . . . 11 (((𝑌 = 𝑥 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) ∧ 𝑦 = 𝑧) → ({𝑥, 𝑧} = {𝑥, 𝑦} ↔ {𝑥, 𝑧} = {𝑥, 𝑧}))
12 eqidd 2759 . . . . . . . . . . 11 ((𝑌 = 𝑥 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → {𝑥, 𝑧} = {𝑥, 𝑧})
138, 11, 12rspcedvd 3546 . . . . . . . . . 10 ((𝑌 = 𝑥 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → ∃𝑦𝑉 {𝑥, 𝑧} = {𝑥, 𝑦})
14 simprr 772 . . . . . . . . . . . 12 ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧})) → (𝐸𝑋) = {𝑥, 𝑧})
15 preq1 4629 . . . . . . . . . . . 12 (𝑌 = 𝑥 → {𝑌, 𝑦} = {𝑥, 𝑦})
1614, 15eqeqan12rd 2777 . . . . . . . . . . 11 ((𝑌 = 𝑥 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → ((𝐸𝑋) = {𝑌, 𝑦} ↔ {𝑥, 𝑧} = {𝑥, 𝑦}))
1716rexbidv 3221 . . . . . . . . . 10 ((𝑌 = 𝑥 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → (∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦} ↔ ∃𝑦𝑉 {𝑥, 𝑧} = {𝑥, 𝑦}))
1813, 17mpbird 260 . . . . . . . . 9 ((𝑌 = 𝑥 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → ∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦})
1918ex 416 . . . . . . . 8 (𝑌 = 𝑥 → ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧})) → ∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦}))
20 simplrl 776 . . . . . . . . . . . 12 ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧})) → 𝑥𝑉)
2120adantl 485 . . . . . . . . . . 11 ((𝑌 = 𝑧 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → 𝑥𝑉)
22 preq2 4630 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → {𝑧, 𝑦} = {𝑧, 𝑥})
2322eqeq2d 2769 . . . . . . . . . . . 12 (𝑦 = 𝑥 → ({𝑥, 𝑧} = {𝑧, 𝑦} ↔ {𝑥, 𝑧} = {𝑧, 𝑥}))
2423adantl 485 . . . . . . . . . . 11 (((𝑌 = 𝑧 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) ∧ 𝑦 = 𝑥) → ({𝑥, 𝑧} = {𝑧, 𝑦} ↔ {𝑥, 𝑧} = {𝑧, 𝑥}))
25 prcom 4628 . . . . . . . . . . . 12 {𝑥, 𝑧} = {𝑧, 𝑥}
2625a1i 11 . . . . . . . . . . 11 ((𝑌 = 𝑧 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → {𝑥, 𝑧} = {𝑧, 𝑥})
2721, 24, 26rspcedvd 3546 . . . . . . . . . 10 ((𝑌 = 𝑧 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → ∃𝑦𝑉 {𝑥, 𝑧} = {𝑧, 𝑦})
28 preq1 4629 . . . . . . . . . . . 12 (𝑌 = 𝑧 → {𝑌, 𝑦} = {𝑧, 𝑦})
2914, 28eqeqan12rd 2777 . . . . . . . . . . 11 ((𝑌 = 𝑧 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → ((𝐸𝑋) = {𝑌, 𝑦} ↔ {𝑥, 𝑧} = {𝑧, 𝑦}))
3029rexbidv 3221 . . . . . . . . . 10 ((𝑌 = 𝑧 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → (∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦} ↔ ∃𝑦𝑉 {𝑥, 𝑧} = {𝑧, 𝑦}))
3127, 30mpbird 260 . . . . . . . . 9 ((𝑌 = 𝑧 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → ∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦})
3231ex 416 . . . . . . . 8 (𝑌 = 𝑧 → ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧})) → ∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦}))
3319, 32jaoi 854 . . . . . . 7 ((𝑌 = 𝑥𝑌 = 𝑧) → ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧})) → ∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦}))
34 elpri 4547 . . . . . . 7 (𝑌 ∈ {𝑥, 𝑧} → (𝑌 = 𝑥𝑌 = 𝑧))
3533, 34syl11 33 . . . . . 6 ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧})) → (𝑌 ∈ {𝑥, 𝑧} → ∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦}))
366, 35sylbid 243 . . . . 5 ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧})) → (𝑌 ∈ (𝐸𝑋) → ∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦}))
3736ex 416 . . . 4 (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) → ((𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}) → (𝑌 ∈ (𝐸𝑋) → ∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦})))
3837rexlimdvva 3218 . . 3 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → (∃𝑥𝑉𝑧𝑉 (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}) → (𝑌 ∈ (𝐸𝑋) → ∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦})))
393, 38mpd 15 . 2 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → (𝑌 ∈ (𝐸𝑋) → ∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦}))
40393impia 1114 1 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸𝑌 ∈ (𝐸𝑋)) → ∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  ∃wrex 3071  {cpr 4527  dom cdm 5527  ‘cfv 6339  Vtxcvtx 26893  iEdgciedg 26894  USGraphcusgr 27046 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464  ax-cnex 10636  ax-resscn 10637  ax-1cn 10638  ax-icn 10639  ax-addcl 10640  ax-addrcl 10641  ax-mulcl 10642  ax-mulrcl 10643  ax-mulcom 10644  ax-addass 10645  ax-mulass 10646  ax-distr 10647  ax-i2m1 10648  ax-1ne0 10649  ax-1rid 10650  ax-rnegex 10651  ax-rrecex 10652  ax-cnre 10653  ax-pre-lttri 10654  ax-pre-lttrn 10655  ax-pre-ltadd 10656  ax-pre-mulgt0 10657 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-tr 5142  df-id 5433  df-eprel 5438  df-po 5446  df-so 5447  df-fr 5486  df-we 5488  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-pred 6130  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7585  df-1st 7698  df-2nd 7699  df-wrecs 7962  df-recs 8023  df-rdg 8061  df-1o 8117  df-2o 8118  df-oadd 8121  df-er 8304  df-en 8533  df-dom 8534  df-sdom 8535  df-fin 8536  df-dju 9368  df-card 9406  df-pnf 10720  df-mnf 10721  df-xr 10722  df-ltxr 10723  df-le 10724  df-sub 10915  df-neg 10916  df-nn 11680  df-2 11742  df-n0 11940  df-z 12026  df-uz 12288  df-fz 12945  df-hash 13746  df-edg 26945  df-umgr 26980  df-usgr 27048 This theorem is referenced by:  usgredgreu  27112
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