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Theorem usgredg4 26387
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 26386 . . 3 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → ∃𝑥𝑉𝑧𝑉 (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))
4 eleq2 2833 . . . . . . . 8 ((𝐸𝑋) = {𝑥, 𝑧} → (𝑌 ∈ (𝐸𝑋) ↔ 𝑌 ∈ {𝑥, 𝑧}))
54adantl 473 . . . . . . 7 ((𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}) → (𝑌 ∈ (𝐸𝑋) ↔ 𝑌 ∈ {𝑥, 𝑧}))
65adantl 473 . . . . . 6 ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧})) → (𝑌 ∈ (𝐸𝑋) ↔ 𝑌 ∈ {𝑥, 𝑧}))
7 simplrr 796 . . . . . . . . . . . 12 ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧})) → 𝑧𝑉)
87adantl 473 . . . . . . . . . . 11 ((𝑌 = 𝑥 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → 𝑧𝑉)
9 preq2 4424 . . . . . . . . . . . . 13 (𝑦 = 𝑧 → {𝑥, 𝑦} = {𝑥, 𝑧})
109eqeq2d 2775 . . . . . . . . . . . 12 (𝑦 = 𝑧 → ({𝑥, 𝑧} = {𝑥, 𝑦} ↔ {𝑥, 𝑧} = {𝑥, 𝑧}))
1110adantl 473 . . . . . . . . . . 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 2766 . . . . . . . . . . 11 ((𝑌 = 𝑥 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → {𝑥, 𝑧} = {𝑥, 𝑧})
138, 11, 12rspcedvd 3468 . . . . . . . . . 10 ((𝑌 = 𝑥 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → ∃𝑦𝑉 {𝑥, 𝑧} = {𝑥, 𝑦})
14 simprr 789 . . . . . . . . . . . 12 ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧})) → (𝐸𝑋) = {𝑥, 𝑧})
15 preq1 4423 . . . . . . . . . . . 12 (𝑌 = 𝑥 → {𝑌, 𝑦} = {𝑥, 𝑦})
1614, 15eqeqan12rd 2783 . . . . . . . . . . 11 ((𝑌 = 𝑥 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → ((𝐸𝑋) = {𝑌, 𝑦} ↔ {𝑥, 𝑧} = {𝑥, 𝑦}))
1716rexbidv 3199 . . . . . . . . . 10 ((𝑌 = 𝑥 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → (∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦} ↔ ∃𝑦𝑉 {𝑥, 𝑧} = {𝑥, 𝑦}))
1813, 17mpbird 248 . . . . . . . . 9 ((𝑌 = 𝑥 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → ∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦})
1918ex 401 . . . . . . . 8 (𝑌 = 𝑥 → ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧})) → ∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦}))
20 simplrl 795 . . . . . . . . . . . 12 ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧})) → 𝑥𝑉)
2120adantl 473 . . . . . . . . . . 11 ((𝑌 = 𝑧 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → 𝑥𝑉)
22 preq2 4424 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → {𝑧, 𝑦} = {𝑧, 𝑥})
2322eqeq2d 2775 . . . . . . . . . . . 12 (𝑦 = 𝑥 → ({𝑥, 𝑧} = {𝑧, 𝑦} ↔ {𝑥, 𝑧} = {𝑧, 𝑥}))
2423adantl 473 . . . . . . . . . . 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 4422 . . . . . . . . . . . 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 3468 . . . . . . . . . 10 ((𝑌 = 𝑧 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → ∃𝑦𝑉 {𝑥, 𝑧} = {𝑧, 𝑦})
28 preq1 4423 . . . . . . . . . . . 12 (𝑌 = 𝑧 → {𝑌, 𝑦} = {𝑧, 𝑦})
2914, 28eqeqan12rd 2783 . . . . . . . . . . 11 ((𝑌 = 𝑧 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → ((𝐸𝑋) = {𝑌, 𝑦} ↔ {𝑥, 𝑧} = {𝑧, 𝑦}))
3029rexbidv 3199 . . . . . . . . . 10 ((𝑌 = 𝑧 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → (∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦} ↔ ∃𝑦𝑉 {𝑥, 𝑧} = {𝑧, 𝑦}))
3127, 30mpbird 248 . . . . . . . . 9 ((𝑌 = 𝑧 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → ∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦})
3231ex 401 . . . . . . . 8 (𝑌 = 𝑧 → ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧})) → ∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦}))
3319, 32jaoi 883 . . . . . . 7 ((𝑌 = 𝑥𝑌 = 𝑧) → ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧})) → ∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦}))
34 elpri 4356 . . . . . . 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 231 . . . . 5 ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧})) → (𝑌 ∈ (𝐸𝑋) → ∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦}))
3736ex 401 . . . 4 (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) → ((𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}) → (𝑌 ∈ (𝐸𝑋) → ∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦})))
3837rexlimdvva 3185 . . 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 1145 1 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸𝑌 ∈ (𝐸𝑋)) → ∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wo 873  w3a 1107   = wceq 1652  wcel 2155  wne 2937  wrex 3056  {cpr 4336  dom cdm 5277  cfv 6068  Vtxcvtx 26165  iEdgciedg 26166  USGraphcusgr 26322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-2o 7765  df-oadd 7768  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-card 9016  df-cda 9243  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-2 11335  df-n0 11539  df-z 11625  df-uz 11887  df-fz 12534  df-hash 13322  df-edg 26217  df-umgr 26255  df-usgr 26324
This theorem is referenced by:  usgredgreu  26388
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