Theorem usgredg4 28474

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 28473	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → ∃𝑥 ∈ 𝑉 ∃𝑧 ∈ 𝑉 (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧}))
4		eleq2 2823	. . . . . . . 8 ⊢ ((𝐸‘𝑋) = {𝑥, 𝑧} → (𝑌 ∈ (𝐸‘𝑋) ↔ 𝑌 ∈ {𝑥, 𝑧}))
5	4	adantl 483	. . . . . . 7 ⊢ ((𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧}) → (𝑌 ∈ (𝐸‘𝑋) ↔ 𝑌 ∈ {𝑥, 𝑧}))
6	5	adantl 483	. . . . . 6 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧})) → (𝑌 ∈ (𝐸‘𝑋) ↔ 𝑌 ∈ {𝑥, 𝑧}))
7		simplrr 777	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧})) → 𝑧 ∈ 𝑉)
8	7	adantl 483	. . . . . . . . . . 11 ⊢ ((𝑌 = 𝑥 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧}))) → 𝑧 ∈ 𝑉)
9		preq2 4739	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → {𝑥, 𝑦} = {𝑥, 𝑧})
10	9	eqeq2d 2744	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → ({𝑥, 𝑧} = {𝑥, 𝑦} ↔ {𝑥, 𝑧} = {𝑥, 𝑧}))
11	10	adantl 483	. . . . . . . . . . 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 2734	. . . . . . . . . . 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 3615	. . . . . . . . . 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 772	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧})) → (𝐸‘𝑋) = {𝑥, 𝑧})
15		preq1 4738	. . . . . . . . . . . 12 ⊢ (𝑌 = 𝑥 → {𝑌, 𝑦} = {𝑥, 𝑦})
16	14, 15	eqeqan12rd 2748	. . . . . . . . . . 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 3179	. . . . . . . . . 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 414	. . . . . . . 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 776	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧})) → 𝑥 ∈ 𝑉)
21	20	adantl 483	. . . . . . . . . . 11 ⊢ ((𝑌 = 𝑧 ∧ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ≠ 𝑧 ∧ (𝐸‘𝑋) = {𝑥, 𝑧}))) → 𝑥 ∈ 𝑉)
22		preq2 4739	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → {𝑧, 𝑦} = {𝑧, 𝑥})
23	22	eqeq2d 2744	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → ({𝑥, 𝑧} = {𝑧, 𝑦} ↔ {𝑥, 𝑧} = {𝑧, 𝑥}))
24	23	adantl 483	. . . . . . . . . . 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 4737	. . . . . . . . . . . 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 3615	. . . . . . . . . 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 4738	. . . . . . . . . . . 12 ⊢ (𝑌 = 𝑧 → {𝑌, 𝑦} = {𝑧, 𝑦})
29	14, 28	eqeqan12rd 2748	. . . . . . . . . . 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 3179	. . . . . . . . . 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 414	. . . . . . . 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 856	. . . . . . 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 4651	. . . . . . 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 239	. . . . 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 414	. . . 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 3212	. . 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 205   ∧ wa 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   ≠ wne 2941  ∃wrex 3071  {cpr 4631  dom cdm 5677  ‘cfv 6544  Vtxcvtx 28256  iEdgciedg 28257  USGraphcusgr 28409

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-2o 8467  df-oadd 8470  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-dju 9896  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-n0 12473  df-z 12559  df-uz 12823  df-fz 13485  df-hash 14291  df-edg 28308  df-umgr 28343  df-usgr 28411

This theorem is referenced by:  usgredgreu  28475