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Theorem usgredg4 26932
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 26931 . . 3 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → ∃𝑥𝑉𝑧𝑉 (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))
4 eleq2 2906 . . . . . . . 8 ((𝐸𝑋) = {𝑥, 𝑧} → (𝑌 ∈ (𝐸𝑋) ↔ 𝑌 ∈ {𝑥, 𝑧}))
54adantl 482 . . . . . . 7 ((𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}) → (𝑌 ∈ (𝐸𝑋) ↔ 𝑌 ∈ {𝑥, 𝑧}))
65adantl 482 . . . . . 6 ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧})) → (𝑌 ∈ (𝐸𝑋) ↔ 𝑌 ∈ {𝑥, 𝑧}))
7 simplrr 774 . . . . . . . . . . . 12 ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧})) → 𝑧𝑉)
87adantl 482 . . . . . . . . . . 11 ((𝑌 = 𝑥 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → 𝑧𝑉)
9 preq2 4669 . . . . . . . . . . . . 13 (𝑦 = 𝑧 → {𝑥, 𝑦} = {𝑥, 𝑧})
109eqeq2d 2837 . . . . . . . . . . . 12 (𝑦 = 𝑧 → ({𝑥, 𝑧} = {𝑥, 𝑦} ↔ {𝑥, 𝑧} = {𝑥, 𝑧}))
1110adantl 482 . . . . . . . . . . 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 2827 . . . . . . . . . . 11 ((𝑌 = 𝑥 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → {𝑥, 𝑧} = {𝑥, 𝑧})
138, 11, 12rspcedvd 3630 . . . . . . . . . 10 ((𝑌 = 𝑥 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → ∃𝑦𝑉 {𝑥, 𝑧} = {𝑥, 𝑦})
14 simprr 769 . . . . . . . . . . . 12 ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧})) → (𝐸𝑋) = {𝑥, 𝑧})
15 preq1 4668 . . . . . . . . . . . 12 (𝑌 = 𝑥 → {𝑌, 𝑦} = {𝑥, 𝑦})
1614, 15eqeqan12rd 2845 . . . . . . . . . . 11 ((𝑌 = 𝑥 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → ((𝐸𝑋) = {𝑌, 𝑦} ↔ {𝑥, 𝑧} = {𝑥, 𝑦}))
1716rexbidv 3302 . . . . . . . . . 10 ((𝑌 = 𝑥 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → (∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦} ↔ ∃𝑦𝑉 {𝑥, 𝑧} = {𝑥, 𝑦}))
1813, 17mpbird 258 . . . . . . . . 9 ((𝑌 = 𝑥 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → ∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦})
1918ex 413 . . . . . . . 8 (𝑌 = 𝑥 → ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧})) → ∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦}))
20 simplrl 773 . . . . . . . . . . . 12 ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧})) → 𝑥𝑉)
2120adantl 482 . . . . . . . . . . 11 ((𝑌 = 𝑧 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → 𝑥𝑉)
22 preq2 4669 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → {𝑧, 𝑦} = {𝑧, 𝑥})
2322eqeq2d 2837 . . . . . . . . . . . 12 (𝑦 = 𝑥 → ({𝑥, 𝑧} = {𝑧, 𝑦} ↔ {𝑥, 𝑧} = {𝑧, 𝑥}))
2423adantl 482 . . . . . . . . . . 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 4667 . . . . . . . . . . . 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 3630 . . . . . . . . . 10 ((𝑌 = 𝑧 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → ∃𝑦𝑉 {𝑥, 𝑧} = {𝑧, 𝑦})
28 preq1 4668 . . . . . . . . . . . 12 (𝑌 = 𝑧 → {𝑌, 𝑦} = {𝑧, 𝑦})
2914, 28eqeqan12rd 2845 . . . . . . . . . . 11 ((𝑌 = 𝑧 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → ((𝐸𝑋) = {𝑌, 𝑦} ↔ {𝑥, 𝑧} = {𝑧, 𝑦}))
3029rexbidv 3302 . . . . . . . . . 10 ((𝑌 = 𝑧 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → (∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦} ↔ ∃𝑦𝑉 {𝑥, 𝑧} = {𝑧, 𝑦}))
3127, 30mpbird 258 . . . . . . . . 9 ((𝑌 = 𝑧 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → ∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦})
3231ex 413 . . . . . . . 8 (𝑌 = 𝑧 → ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧})) → ∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦}))
3319, 32jaoi 853 . . . . . . 7 ((𝑌 = 𝑥𝑌 = 𝑧) → ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧})) → ∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦}))
34 elpri 4586 . . . . . . 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 241 . . . . 5 ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧})) → (𝑌 ∈ (𝐸𝑋) → ∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦}))
3736ex 413 . . . 4 (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) → ((𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}) → (𝑌 ∈ (𝐸𝑋) → ∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦})))
3837rexlimdvva 3299 . . 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 1111 1 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸𝑌 ∈ (𝐸𝑋)) → ∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wo 843  w3a 1081   = wceq 1530  wcel 2107  wne 3021  wrex 3144  {cpr 4566  dom cdm 5554  cfv 6354  Vtxcvtx 26714  iEdgciedg 26715  USGraphcusgr 26867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7574  df-1st 7685  df-2nd 7686  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-2o 8099  df-oadd 8102  df-er 8284  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-dju 9324  df-card 9362  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-2 11694  df-n0 11892  df-z 11976  df-uz 12238  df-fz 12888  df-hash 13686  df-edg 26766  df-umgr 26801  df-usgr 26869
This theorem is referenced by:  usgredgreu  26933
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