Theorem usgrarnedg 25913

Description: For each edge in a simple graph, there are two distinct vertices which are connected by this edge. (Contributed by Alexander van der Vekens, 9-Dec-2017.)

Assertion

Ref	Expression
usgrarnedg	⊢ ((𝑉 USGrph 𝐸 ∧ 𝑌 ∈ ran 𝐸) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑌 = {𝑎, 𝑏}))

Distinct variable groups:   𝐸,𝑎,𝑏   𝑌,𝑎,𝑏   𝑉,𝑎,𝑏

Proof of Theorem usgrarnedg

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		usgrafun 25878	. . . 4 ⊢ (𝑉 USGrph 𝐸 → Fun 𝐸)
2		funfn 5833	. . . . 5 ⊢ (Fun 𝐸 ↔ 𝐸 Fn dom 𝐸)
3	2	biimpi 205	. . . 4 ⊢ (Fun 𝐸 → 𝐸 Fn dom 𝐸)
4		fvelrnb 6153	. . . 4 ⊢ (𝐸 Fn dom 𝐸 → (𝑌 ∈ ran 𝐸 ↔ ∃𝑥 ∈ dom 𝐸(𝐸‘𝑥) = 𝑌))
5	1, 3, 4	3syl 18	. . 3 ⊢ (𝑉 USGrph 𝐸 → (𝑌 ∈ ran 𝐸 ↔ ∃𝑥 ∈ dom 𝐸(𝐸‘𝑥) = 𝑌))
6		usgraf0 25877	. . . . 5 ⊢ (𝑉 USGrph 𝐸 → 𝐸:dom 𝐸–1-1→{𝑦 ∈ 𝒫 𝑉 ∣ (#‘𝑦) = 2})
7		f1f 6014	. . . . 5 ⊢ (𝐸:dom 𝐸–1-1→{𝑦 ∈ 𝒫 𝑉 ∣ (#‘𝑦) = 2} → 𝐸:dom 𝐸⟶{𝑦 ∈ 𝒫 𝑉 ∣ (#‘𝑦) = 2})
8		ffvelrn 6265	. . . . . . 7 ⊢ ((𝐸:dom 𝐸⟶{𝑦 ∈ 𝒫 𝑉 ∣ (#‘𝑦) = 2} ∧ 𝑥 ∈ dom 𝐸) → (𝐸‘𝑥) ∈ {𝑦 ∈ 𝒫 𝑉 ∣ (#‘𝑦) = 2})
9		eleq1 2676	. . . . . . . 8 ⊢ ((𝐸‘𝑥) = 𝑌 → ((𝐸‘𝑥) ∈ {𝑦 ∈ 𝒫 𝑉 ∣ (#‘𝑦) = 2} ↔ 𝑌 ∈ {𝑦 ∈ 𝒫 𝑉 ∣ (#‘𝑦) = 2}))
10		fveq2 6103	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑌 → (#‘𝑦) = (#‘𝑌))
11	10	eqeq1d 2612	. . . . . . . . . 10 ⊢ (𝑦 = 𝑌 → ((#‘𝑦) = 2 ↔ (#‘𝑌) = 2))
12	11	elrab 3331	. . . . . . . . 9 ⊢ (𝑌 ∈ {𝑦 ∈ 𝒫 𝑉 ∣ (#‘𝑦) = 2} ↔ (𝑌 ∈ 𝒫 𝑉 ∧ (#‘𝑌) = 2))
13		hash2prde 13109	. . . . . . . . . 10 ⊢ ((𝑌 ∈ 𝒫 𝑉 ∧ (#‘𝑌) = 2) → ∃𝑎∃𝑏(𝑎 ≠ 𝑏 ∧ 𝑌 = {𝑎, 𝑏}))
14		eleq1 2676	. . . . . . . . . . . . . . . . 17 ⊢ (𝑌 = {𝑎, 𝑏} → (𝑌 ∈ 𝒫 𝑉 ↔ {𝑎, 𝑏} ∈ 𝒫 𝑉))
15		prex 4836	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝑎, 𝑏} ∈ V
16	15	elpw 4114	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝑎, 𝑏} ∈ 𝒫 𝑉 ↔ {𝑎, 𝑏} ⊆ 𝑉)
17		vex 3176	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑎 ∈ V
18		vex 3176	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑏 ∈ V
19	17, 18	prss 4291	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ↔ {𝑎, 𝑏} ⊆ 𝑉)
20	16, 19	sylbb2 227	. . . . . . . . . . . . . . . . 17 ⊢ ({𝑎, 𝑏} ∈ 𝒫 𝑉 → (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))
21	14, 20	syl6bi 242	. . . . . . . . . . . . . . . 16 ⊢ (𝑌 = {𝑎, 𝑏} → (𝑌 ∈ 𝒫 𝑉 → (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)))
22	21	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑌 = {𝑎, 𝑏}) → (𝑌 ∈ 𝒫 𝑉 → (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)))
23	22	imdistanri 723	. . . . . . . . . . . . . 14 ⊢ ((𝑌 ∈ 𝒫 𝑉 ∧ (𝑎 ≠ 𝑏 ∧ 𝑌 = {𝑎, 𝑏})) → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ≠ 𝑏 ∧ 𝑌 = {𝑎, 𝑏})))
24	23	ex 449	. . . . . . . . . . . . 13 ⊢ (𝑌 ∈ 𝒫 𝑉 → ((𝑎 ≠ 𝑏 ∧ 𝑌 = {𝑎, 𝑏}) → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ≠ 𝑏 ∧ 𝑌 = {𝑎, 𝑏}))))
25	24	2eximdv 1835	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ 𝒫 𝑉 → (∃𝑎∃𝑏(𝑎 ≠ 𝑏 ∧ 𝑌 = {𝑎, 𝑏}) → ∃𝑎∃𝑏((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ≠ 𝑏 ∧ 𝑌 = {𝑎, 𝑏}))))
26		r2ex 3043	. . . . . . . . . . . 12 ⊢ (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑌 = {𝑎, 𝑏}) ↔ ∃𝑎∃𝑏((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ≠ 𝑏 ∧ 𝑌 = {𝑎, 𝑏})))
27	25, 26	syl6ibr 241	. . . . . . . . . . 11 ⊢ (𝑌 ∈ 𝒫 𝑉 → (∃𝑎∃𝑏(𝑎 ≠ 𝑏 ∧ 𝑌 = {𝑎, 𝑏}) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑌 = {𝑎, 𝑏})))
28	27	adantr 480	. . . . . . . . . 10 ⊢ ((𝑌 ∈ 𝒫 𝑉 ∧ (#‘𝑌) = 2) → (∃𝑎∃𝑏(𝑎 ≠ 𝑏 ∧ 𝑌 = {𝑎, 𝑏}) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑌 = {𝑎, 𝑏})))
29	13, 28	mpd 15	. . . . . . . . 9 ⊢ ((𝑌 ∈ 𝒫 𝑉 ∧ (#‘𝑌) = 2) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑌 = {𝑎, 𝑏}))
30	12, 29	sylbi 206	. . . . . . . 8 ⊢ (𝑌 ∈ {𝑦 ∈ 𝒫 𝑉 ∣ (#‘𝑦) = 2} → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑌 = {𝑎, 𝑏}))
31	9, 30	syl6bi 242	. . . . . . 7 ⊢ ((𝐸‘𝑥) = 𝑌 → ((𝐸‘𝑥) ∈ {𝑦 ∈ 𝒫 𝑉 ∣ (#‘𝑦) = 2} → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑌 = {𝑎, 𝑏})))
32	8, 31	syl5com 31	. . . . . 6 ⊢ ((𝐸:dom 𝐸⟶{𝑦 ∈ 𝒫 𝑉 ∣ (#‘𝑦) = 2} ∧ 𝑥 ∈ dom 𝐸) → ((𝐸‘𝑥) = 𝑌 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑌 = {𝑎, 𝑏})))
33	32	ex 449	. . . . 5 ⊢ (𝐸:dom 𝐸⟶{𝑦 ∈ 𝒫 𝑉 ∣ (#‘𝑦) = 2} → (𝑥 ∈ dom 𝐸 → ((𝐸‘𝑥) = 𝑌 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑌 = {𝑎, 𝑏}))))
34	6, 7, 33	3syl 18	. . . 4 ⊢ (𝑉 USGrph 𝐸 → (𝑥 ∈ dom 𝐸 → ((𝐸‘𝑥) = 𝑌 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑌 = {𝑎, 𝑏}))))
35	34	rexlimdv 3012	. . 3 ⊢ (𝑉 USGrph 𝐸 → (∃𝑥 ∈ dom 𝐸(𝐸‘𝑥) = 𝑌 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑌 = {𝑎, 𝑏})))
36	5, 35	sylbid 229	. 2 ⊢ (𝑉 USGrph 𝐸 → (𝑌 ∈ ran 𝐸 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑌 = {𝑎, 𝑏})))
37	36	imp 444	1 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑌 ∈ ran 𝐸) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑌 = {𝑎, 𝑏}))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  {crab 2900   ⊆ wss 3540  𝒫 cpw 4108  {cpr 4127   class class class wbr 4583  dom cdm 5038  ran crn 5039  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804  2c2 10947  #chash 12979   USGrph cusg 25859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-usgra 25862

This theorem is referenced by:  edgprvtx  25914  usgraedg3  25915  usgrarnedg1  25918  usgrasscusgra  26011  sizeusglecusglem1  26012