Theorem usgredg4 29308

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 29307	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → ∃𝑥 ∈ 𝑉 ∃𝑧 ∈ 𝑉 (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧}))
4		eleq2 2826	. . . . . . . 8 ⊢ ((𝐸‘𝑋) = {𝑥, 𝑧} → (𝑌 ∈ (𝐸‘𝑋) ↔ 𝑌 ∈ {𝑥, 𝑧}))
5	4	adantl 481	. . . . . . 7 ⊢ ((𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧}) → (𝑌 ∈ (𝐸‘𝑋) ↔ 𝑌 ∈ {𝑥, 𝑧}))
6	5	adantl 481	. . . . . 6 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧})) → (𝑌 ∈ (𝐸‘𝑋) ↔ 𝑌 ∈ {𝑥, 𝑧}))
7		simplrr 778	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧})) → 𝑧 ∈ 𝑉)
8	7	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑌 = 𝑥 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧}))) → 𝑧 ∈ 𝑉)
9		preq2 4693	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → {𝑥, 𝑦} = {𝑥, 𝑧})
10	9	eqeq2d 2748	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → ({𝑥, 𝑧} = {𝑥, 𝑦} ↔ {𝑥, 𝑧} = {𝑥, 𝑧}))
11	10	adantl 481	. . . . . . . . . . 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 2738	. . . . . . . . . . 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 3580	. . . . . . . . . 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 773	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧})) → (𝐸‘𝑋) = {𝑥, 𝑧})
15		preq1 4692	. . . . . . . . . . . 12 ⊢ (𝑌 = 𝑥 → {𝑌, 𝑦} = {𝑥, 𝑦})
16	14, 15	eqeqan12rd 2752	. . . . . . . . . . 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 3162	. . . . . . . . . 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 257	. . . . . . . . 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 412	. . . . . . . 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 777	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧})) → 𝑥 ∈ 𝑉)
21	20	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑌 = 𝑧 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧}))) → 𝑥 ∈ 𝑉)
22		preq2 4693	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → {𝑧, 𝑦} = {𝑧, 𝑥})
23	22	eqeq2d 2748	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → ({𝑥, 𝑧} = {𝑧, 𝑦} ↔ {𝑥, 𝑧} = {𝑧, 𝑥}))
24	23	adantl 481	. . . . . . . . . . 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 4691	. . . . . . . . . . . 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 3580	. . . . . . . . . 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 4692	. . . . . . . . . . . 12 ⊢ (𝑌 = 𝑧 → {𝑌, 𝑦} = {𝑧, 𝑦})
29	14, 28	eqeqan12rd 2752	. . . . . . . . . . 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 3162	. . . . . . . . . 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 257	. . . . . . . . 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 412	. . . . . . . 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 858	. . . . . . 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 4606	. . . . . . 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 240	. . . . 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 412	. . . 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 3195	. . 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 1118	1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸 ∧ 𝑌 ∈ (𝐸‘𝑋)) → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∃wrex 3062  {cpr 4584  dom cdm 5634  ‘cfv 6502  Vtxcvtx 29087  iEdgciedg 29088  USGraphcusgr 29240

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692  ax-cnex 11096  ax-resscn 11097  ax-1cn 11098  ax-icn 11099  ax-addcl 11100  ax-addrcl 11101  ax-mulcl 11102  ax-mulrcl 11103  ax-mulcom 11104  ax-addass 11105  ax-mulass 11106  ax-distr 11107  ax-i2m1 11108  ax-1ne0 11109  ax-1rid 11110  ax-rnegex 11111  ax-rrecex 11112  ax-cnre 11113  ax-pre-lttri 11114  ax-pre-lttrn 11115  ax-pre-ltadd 11116  ax-pre-mulgt0 11117

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-riota 7327  df-ov 7373  df-oprab 7374  df-mpo 7375  df-om 7821  df-1st 7945  df-2nd 7946  df-frecs 8235  df-wrecs 8266  df-recs 8315  df-rdg 8353  df-1o 8409  df-2o 8410  df-oadd 8413  df-er 8647  df-en 8898  df-dom 8899  df-sdom 8900  df-fin 8901  df-dju 9827  df-card 9865  df-pnf 11182  df-mnf 11183  df-xr 11184  df-ltxr 11185  df-le 11186  df-sub 11380  df-neg 11381  df-nn 12160  df-2 12222  df-n0 12416  df-z 12503  df-uz 12766  df-fz 13438  df-hash 14268  df-edg 29139  df-umgr 29174  df-usgr 29242

This theorem is referenced by:  usgredgreu  29309