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Theorem usgredg4 26329
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 26328 . . 3 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → ∃𝑥𝑉𝑧𝑉 (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))
4 eleq2 2828 . . . . . . . 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 820 . . . . . . . . . . . 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 4413 . . . . . . . . . . . . 13 (𝑦 = 𝑧 → {𝑥, 𝑦} = {𝑥, 𝑧})
109eqeq2d 2770 . . . . . . . . . . . 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 2761 . . . . . . . . . . 11 ((𝑌 = 𝑥 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → {𝑥, 𝑧} = {𝑥, 𝑧})
138, 11, 12rspcedvd 3456 . . . . . . . . . 10 ((𝑌 = 𝑥 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → ∃𝑦𝑉 {𝑥, 𝑧} = {𝑥, 𝑦})
14 simprr 813 . . . . . . . . . . . 12 ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧})) → (𝐸𝑋) = {𝑥, 𝑧})
15 preq1 4412 . . . . . . . . . . . 12 (𝑌 = 𝑥 → {𝑌, 𝑦} = {𝑥, 𝑦})
1614, 15eqeqan12rd 2778 . . . . . . . . . . 11 ((𝑌 = 𝑥 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → ((𝐸𝑋) = {𝑌, 𝑦} ↔ {𝑥, 𝑧} = {𝑥, 𝑦}))
1716rexbidv 3190 . . . . . . . . . 10 ((𝑌 = 𝑥 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → (∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦} ↔ ∃𝑦𝑉 {𝑥, 𝑧} = {𝑥, 𝑦}))
1813, 17mpbird 247 . . . . . . . . 9 ((𝑌 = 𝑥 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → ∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦})
1918ex 449 . . . . . . . 8 (𝑌 = 𝑥 → ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧})) → ∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦}))
20 simplrl 819 . . . . . . . . . . . 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 4413 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → {𝑧, 𝑦} = {𝑧, 𝑥})
2322eqeq2d 2770 . . . . . . . . . . . 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 4411 . . . . . . . . . . . 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 3456 . . . . . . . . . 10 ((𝑌 = 𝑧 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → ∃𝑦𝑉 {𝑥, 𝑧} = {𝑧, 𝑦})
28 preq1 4412 . . . . . . . . . . . 12 (𝑌 = 𝑧 → {𝑌, 𝑦} = {𝑧, 𝑦})
2914, 28eqeqan12rd 2778 . . . . . . . . . . 11 ((𝑌 = 𝑧 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → ((𝐸𝑋) = {𝑌, 𝑦} ↔ {𝑥, 𝑧} = {𝑧, 𝑦}))
3029rexbidv 3190 . . . . . . . . . 10 ((𝑌 = 𝑧 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → (∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦} ↔ ∃𝑦𝑉 {𝑥, 𝑧} = {𝑧, 𝑦}))
3127, 30mpbird 247 . . . . . . . . 9 ((𝑌 = 𝑧 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}))) → ∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦})
3231ex 449 . . . . . . . 8 (𝑌 = 𝑧 → ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧})) → ∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦}))
3319, 32jaoi 393 . . . . . . 7 ((𝑌 = 𝑥𝑌 = 𝑧) → ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧})) → ∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦}))
34 elpri 4342 . . . . . . 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 230 . . . . 5 ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) ∧ (𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧})) → (𝑌 ∈ (𝐸𝑋) → ∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦}))
3736ex 449 . . . 4 (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥𝑉𝑧𝑉)) → ((𝑥𝑧 ∧ (𝐸𝑋) = {𝑥, 𝑧}) → (𝑌 ∈ (𝐸𝑋) → ∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦})))
3837rexlimdvva 3176 . . 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 1110 1 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸𝑌 ∈ (𝐸𝑋)) → ∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932  wrex 3051  {cpr 4323  dom cdm 5266  cfv 6049  Vtxcvtx 26094  iEdgciedg 26095  USGraphcusgr 26264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-2o 7731  df-oadd 7734  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-card 8975  df-cda 9202  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-nn 11233  df-2 11291  df-n0 11505  df-z 11590  df-uz 11900  df-fz 12540  df-hash 13332  df-edg 26160  df-umgr 26198  df-usgr 26266
This theorem is referenced by:  usgredgreu  26330
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