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Theorem usgredg4 29249
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 29248 . . 3 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → ∃𝑥𝑉𝑧𝑉 (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))
4 eleq2 2828 . . . . . . . 8 ((𝐸𝑋) = {𝑥, 𝑧} → (𝑌 ∈ (𝐸𝑋) ↔ 𝑌 ∈ {𝑥, 𝑧}))
54adantl 481 . . . . . . 7 ((𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}) → (𝑌 ∈ (𝐸𝑋) ↔ 𝑌 ∈ {𝑥, 𝑧}))
65adantl 481 . . . . . 6 ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧})) → (𝑌 ∈ (𝐸𝑋) ↔ 𝑌 ∈ {𝑥, 𝑧}))
7 simplrr 778 . . . . . . . . . . . 12 ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧})) → 𝑧𝑉)
87adantl 481 . . . . . . . . . . 11 ((𝑌 = 𝑥 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → 𝑧𝑉)
9 preq2 4739 . . . . . . . . . . . . 13 (𝑦 = 𝑧 → {𝑥, 𝑦} = {𝑥, 𝑧})
109eqeq2d 2746 . . . . . . . . . . . 12 (𝑦 = 𝑧 → ({𝑥, 𝑧} = {𝑥, 𝑦} ↔ {𝑥, 𝑧} = {𝑥, 𝑧}))
1110adantl 481 . . . . . . . . . . 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 2736 . . . . . . . . . . 11 ((𝑌 = 𝑥 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → {𝑥, 𝑧} = {𝑥, 𝑧})
138, 11, 12rspcedvd 3624 . . . . . . . . . 10 ((𝑌 = 𝑥 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → ∃𝑦𝑉 {𝑥, 𝑧} = {𝑥, 𝑦})
14 simprr 773 . . . . . . . . . . . 12 ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧})) → (𝐸𝑋) = {𝑥, 𝑧})
15 preq1 4738 . . . . . . . . . . . 12 (𝑌 = 𝑥 → {𝑌, 𝑦} = {𝑥, 𝑦})
1614, 15eqeqan12rd 2750 . . . . . . . . . . 11 ((𝑌 = 𝑥 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → ((𝐸𝑋) = {𝑌, 𝑦} ↔ {𝑥, 𝑧} = {𝑥, 𝑦}))
1716rexbidv 3177 . . . . . . . . . 10 ((𝑌 = 𝑥 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → (∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦} ↔ ∃𝑦𝑉 {𝑥, 𝑧} = {𝑥, 𝑦}))
1813, 17mpbird 257 . . . . . . . . 9 ((𝑌 = 𝑥 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → ∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦})
1918ex 412 . . . . . . . 8 (𝑌 = 𝑥 → ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧})) → ∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦}))
20 simplrl 777 . . . . . . . . . . . 12 ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧})) → 𝑥𝑉)
2120adantl 481 . . . . . . . . . . 11 ((𝑌 = 𝑧 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → 𝑥𝑉)
22 preq2 4739 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → {𝑧, 𝑦} = {𝑧, 𝑥})
2322eqeq2d 2746 . . . . . . . . . . . 12 (𝑦 = 𝑥 → ({𝑥, 𝑧} = {𝑧, 𝑦} ↔ {𝑥, 𝑧} = {𝑧, 𝑥}))
2423adantl 481 . . . . . . . . . . 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 4737 . . . . . . . . . . . 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 3624 . . . . . . . . . 10 ((𝑌 = 𝑧 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → ∃𝑦𝑉 {𝑥, 𝑧} = {𝑧, 𝑦})
28 preq1 4738 . . . . . . . . . . . 12 (𝑌 = 𝑧 → {𝑌, 𝑦} = {𝑧, 𝑦})
2914, 28eqeqan12rd 2750 . . . . . . . . . . 11 ((𝑌 = 𝑧 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → ((𝐸𝑋) = {𝑌, 𝑦} ↔ {𝑥, 𝑧} = {𝑧, 𝑦}))
3029rexbidv 3177 . . . . . . . . . 10 ((𝑌 = 𝑧 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → (∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦} ↔ ∃𝑦𝑉 {𝑥, 𝑧} = {𝑧, 𝑦}))
3127, 30mpbird 257 . . . . . . . . 9 ((𝑌 = 𝑧 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → ∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦})
3231ex 412 . . . . . . . 8 (𝑌 = 𝑧 → ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧})) → ∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦}))
3319, 32jaoi 857 . . . . . . 7 ((𝑌 = 𝑥𝑌 = 𝑧) → ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧})) → ∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦}))
34 elpri 4654 . . . . . . 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 240 . . . . 5 ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧})) → (𝑌 ∈ (𝐸𝑋) → ∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦}))
3736ex 412 . . . 4 (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) → ((𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}) → (𝑌 ∈ (𝐸𝑋) → ∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦})))
3837rexlimdvva 3211 . . 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 1116 1 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸𝑌 ∈ (𝐸𝑋)) → ∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wrex 3068  {cpr 4633  dom cdm 5689  cfv 6563  Vtxcvtx 29028  iEdgciedg 29029  USGraphcusgr 29181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-dju 9939  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-hash 14367  df-edg 29080  df-umgr 29115  df-usgr 29183
This theorem is referenced by:  usgredgreu  29250
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