Theorem usgredg4 29420

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 𝐸 ∧ 𝑌 ∈ (𝐸‘𝑋)) → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {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.

Step	Hyp	Ref	Expression
1		usgredg3.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2		usgredg3.e	. . . 4 ⊢ 𝐸 = (iEdg‘𝐺)
3	1, 2	usgredg3 29419	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → ∃𝑥 ∈ 𝑉 ∃𝑧 ∈ 𝑉 (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧}))
4		eleq2 2853	. . . . . . . 8 ⊢ ((𝐸‘𝑋) = {𝑥, 𝑧} → (𝑌 ∈ (𝐸‘𝑋) ↔ 𝑌 ∈ {𝑥, 𝑧}))
5	4	adantl 485	. . . . . . 7 ⊢ ((𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧}) → (𝑌 ∈ (𝐸‘𝑋) ↔ 𝑌 ∈ {𝑥, 𝑧}))
6	5	adantl 485	. . . . . 6 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧})) → (𝑌 ∈ (𝐸‘𝑋) ↔ 𝑌 ∈ {𝑥, 𝑧}))
7		simplrr 787	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧})) → 𝑧 ∈ 𝑉)
8	7	adantl 485	. . . . . . . . . . 11 ⊢ ((𝑌 = 𝑥 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧}))) → 𝑧 ∈ 𝑉)
9		preq2 4695	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → {𝑥, 𝑦} = {𝑥, 𝑧})
10	9	eqeq2d 2775	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → ({𝑥, 𝑧} = {𝑥, 𝑦} ↔ {𝑥, 𝑧} = {𝑥, 𝑧}))
11	10	adantl 485	. . . . . . . . . . 11 ⊢ (((𝑌 = 𝑥 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧}))) ∧ 𝑦 = 𝑧) → ({𝑥, 𝑧} = {𝑥, 𝑦} ↔ {𝑥, 𝑧} = {𝑥, 𝑧}))
12		eqidd 2765	. . . . . . . . . . 11 ⊢ ((𝑌 = 𝑥 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧}))) → {𝑥, 𝑧} = {𝑥, 𝑧})
13	8, 11, 12	rspcedvd 3585	. . . . . . . . . 10 ⊢ ((𝑌 = 𝑥 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧}))) → ∃𝑦 ∈ 𝑉 {𝑥, 𝑧} = {𝑥, 𝑦})
14		simprr 782	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧})) → (𝐸‘𝑋) = {𝑥, 𝑧})
15		preq1 4694	. . . . . . . . . . . 12 ⊢ (𝑌 = 𝑥 → {𝑌, 𝑦} = {𝑥, 𝑦})
16	14, 15	eqeqan12rd 2779	. . . . . . . . . . 11 ⊢ ((𝑌 = 𝑥 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧}))) → ((𝐸‘𝑋) = {𝑌, 𝑦} ↔ {𝑥, 𝑧} = {𝑥, 𝑦}))
17	16	rexbidv 3188	. . . . . . . . . 10 ⊢ ((𝑌 = 𝑥 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧}))) → (∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦} ↔ ∃𝑦 ∈ 𝑉 {𝑥, 𝑧} = {𝑥, 𝑦}))
18	13, 17	mpbird 259	. . . . . . . . 9 ⊢ ((𝑌 = 𝑥 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧}))) → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦})
19	18	ex 416	. . . . . . . 8 ⊢ (𝑌 = 𝑥 → ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧})) → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦}))
20		simplrl 786	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧})) → 𝑥 ∈ 𝑉)
21	20	adantl 485	. . . . . . . . . . 11 ⊢ ((𝑌 = 𝑧 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧}))) → 𝑥 ∈ 𝑉)
22		preq2 4695	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → {𝑧, 𝑦} = {𝑧, 𝑥})
23	22	eqeq2d 2775	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → ({𝑥, 𝑧} = {𝑧, 𝑦} ↔ {𝑥, 𝑧} = {𝑧, 𝑥}))
24	23	adantl 485	. . . . . . . . . . 11 ⊢ (((𝑌 = 𝑧 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧}))) ∧ 𝑦 = 𝑥) → ({𝑥, 𝑧} = {𝑧, 𝑦} ↔ {𝑥, 𝑧} = {𝑧, 𝑥}))
25		prcom 4693	. . . . . . . . . . . 12 ⊢ {𝑥, 𝑧} = {𝑧, 𝑥}
26	25	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑌 = 𝑧 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧}))) → {𝑥, 𝑧} = {𝑧, 𝑥})
27	21, 24, 26	rspcedvd 3585	. . . . . . . . . 10 ⊢ ((𝑌 = 𝑧 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧}))) → ∃𝑦 ∈ 𝑉 {𝑥, 𝑧} = {𝑧, 𝑦})
28		preq1 4694	. . . . . . . . . . . 12 ⊢ (𝑌 = 𝑧 → {𝑌, 𝑦} = {𝑧, 𝑦})
29	14, 28	eqeqan12rd 2779	. . . . . . . . . . 11 ⊢ ((𝑌 = 𝑧 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧}))) → ((𝐸‘𝑋) = {𝑌, 𝑦} ↔ {𝑥, 𝑧} = {𝑧, 𝑦}))
30	29	rexbidv 3188	. . . . . . . . . 10 ⊢ ((𝑌 = 𝑧 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧}))) → (∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦} ↔ ∃𝑦 ∈ 𝑉 {𝑥, 𝑧} = {𝑧, 𝑦}))
31	27, 30	mpbird 259	. . . . . . . . 9 ⊢ ((𝑌 = 𝑧 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧}))) → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦})
32	31	ex 416	. . . . . . . 8 ⊢ (𝑌 = 𝑧 → ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧})) → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦}))
33	19, 32	jaoi 868	. . . . . . 7 ⊢ ((𝑌 = 𝑥 ∨ 𝑌 = 𝑧) → ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧})) → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦}))
34		elpri 4608	. . . . . . 7 ⊢ (𝑌 ∈ {𝑥, 𝑧} → (𝑌 = 𝑥 ∨ 𝑌 = 𝑧))
35	33, 34	syl11 33	. . . . . 6 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧})) → (𝑌 ∈ {𝑥, 𝑧} → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦}))
36	6, 35	sylbid 242	. . . . 5 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧})) → (𝑌 ∈ (𝐸‘𝑋) → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦}))
37	36	ex 416	. . . 4 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧}) → (𝑌 ∈ (𝐸‘𝑋) → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦})))
38	37	rexlimdvva 3221	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → (∃𝑥 ∈ 𝑉 ∃𝑧 ∈ 𝑉 (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧}) → (𝑌 ∈ (𝐸‘𝑋) → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦})))
39	3, 38	mpd 15	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → (𝑌 ∈ (𝐸‘𝑋) → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦}))
40	39	3impia 1131	1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸 ∧ 𝑌 ∈ (𝐸‘𝑋)) → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦})

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144   ≠ wne 2959  ∃wrex 3088  {cpr 4586  dom cdm 5649  ‘cfv 6523  Vtxcvtx 29199  iEdgciedg 29200  USGraphcusgr 29352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3064  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-om 7849  df-1st 7972  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-1o 8439  df-2o 8440  df-oadd 8443  df-er 8680  df-en 8930  df-dom 8931  df-sdom 8932  df-fin 8933  df-dju 9861  df-card 9899  df-pnf 11220  df-mnf 11221  df-xr 11222  df-ltxr 11223  df-le 11224  df-sub 11418  df-neg 11419  df-nn 12213  df-2 12282  df-n0 12484  df-z 12571  df-uz 12842  df-fz 13515  df-hash 14346  df-edg 29251  df-umgr 29286  df-usgr 29354

This theorem is referenced by:  usgredgreu  29421