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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 𝐸𝑌 ∈ ran 𝐸) → ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑌 = {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.
StepHypRef Expression
1 usgrafun 25878 . . . 4 (𝑉 USGrph 𝐸 → Fun 𝐸)
2 funfn 5833 . . . . 5 (Fun 𝐸𝐸 Fn dom 𝐸)
32biimpi 205 . . . 4 (Fun 𝐸𝐸 Fn dom 𝐸)
4 fvelrnb 6153 . . . 4 (𝐸 Fn dom 𝐸 → (𝑌 ∈ ran 𝐸 ↔ ∃𝑥 ∈ dom 𝐸(𝐸𝑥) = 𝑌))
51, 3, 43syl 18 . . 3 (𝑉 USGrph 𝐸 → (𝑌 ∈ ran 𝐸 ↔ ∃𝑥 ∈ dom 𝐸(𝐸𝑥) = 𝑌))
6 usgraf0 25877 . . . . 5 (𝑉 USGrph 𝐸𝐸:dom 𝐸1-1→{𝑦 ∈ 𝒫 𝑉 ∣ (#‘𝑦) = 2})
7 f1f 6014 . . . . 5 (𝐸:dom 𝐸1-1→{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) → (𝐸𝑥) ∈ {𝑦 ∈ 𝒫 𝑉 ∣ (#‘𝑦) = 2})
9 eleq1 2676 . . . . . . . 8 ((𝐸𝑥) = 𝑌 → ((𝐸𝑥) ∈ {𝑦 ∈ 𝒫 𝑉 ∣ (#‘𝑦) = 2} ↔ 𝑌 ∈ {𝑦 ∈ 𝒫 𝑉 ∣ (#‘𝑦) = 2}))
10 fveq2 6103 . . . . . . . . . . 11 (𝑦 = 𝑌 → (#‘𝑦) = (#‘𝑌))
1110eqeq1d 2612 . . . . . . . . . 10 (𝑦 = 𝑌 → ((#‘𝑦) = 2 ↔ (#‘𝑌) = 2))
1211elrab 3331 . . . . . . . . 9 (𝑌 ∈ {𝑦 ∈ 𝒫 𝑉 ∣ (#‘𝑦) = 2} ↔ (𝑌 ∈ 𝒫 𝑉 ∧ (#‘𝑌) = 2))
13 hash2prde 13109 . . . . . . . . . 10 ((𝑌 ∈ 𝒫 𝑉 ∧ (#‘𝑌) = 2) → ∃𝑎𝑏(𝑎𝑏𝑌 = {𝑎, 𝑏}))
14 eleq1 2676 . . . . . . . . . . . . . . . . 17 (𝑌 = {𝑎, 𝑏} → (𝑌 ∈ 𝒫 𝑉 ↔ {𝑎, 𝑏} ∈ 𝒫 𝑉))
15 prex 4836 . . . . . . . . . . . . . . . . . . 19 {𝑎, 𝑏} ∈ V
1615elpw 4114 . . . . . . . . . . . . . . . . . 18 ({𝑎, 𝑏} ∈ 𝒫 𝑉 ↔ {𝑎, 𝑏} ⊆ 𝑉)
17 vex 3176 . . . . . . . . . . . . . . . . . . 19 𝑎 ∈ V
18 vex 3176 . . . . . . . . . . . . . . . . . . 19 𝑏 ∈ V
1917, 18prss 4291 . . . . . . . . . . . . . . . . . 18 ((𝑎𝑉𝑏𝑉) ↔ {𝑎, 𝑏} ⊆ 𝑉)
2016, 19sylbb2 227 . . . . . . . . . . . . . . . . 17 ({𝑎, 𝑏} ∈ 𝒫 𝑉 → (𝑎𝑉𝑏𝑉))
2114, 20syl6bi 242 . . . . . . . . . . . . . . . 16 (𝑌 = {𝑎, 𝑏} → (𝑌 ∈ 𝒫 𝑉 → (𝑎𝑉𝑏𝑉)))
2221adantl 481 . . . . . . . . . . . . . . 15 ((𝑎𝑏𝑌 = {𝑎, 𝑏}) → (𝑌 ∈ 𝒫 𝑉 → (𝑎𝑉𝑏𝑉)))
2322imdistanri 723 . . . . . . . . . . . . . 14 ((𝑌 ∈ 𝒫 𝑉 ∧ (𝑎𝑏𝑌 = {𝑎, 𝑏})) → ((𝑎𝑉𝑏𝑉) ∧ (𝑎𝑏𝑌 = {𝑎, 𝑏})))
2423ex 449 . . . . . . . . . . . . 13 (𝑌 ∈ 𝒫 𝑉 → ((𝑎𝑏𝑌 = {𝑎, 𝑏}) → ((𝑎𝑉𝑏𝑉) ∧ (𝑎𝑏𝑌 = {𝑎, 𝑏}))))
25242eximdv 1835 . . . . . . . . . . . 12 (𝑌 ∈ 𝒫 𝑉 → (∃𝑎𝑏(𝑎𝑏𝑌 = {𝑎, 𝑏}) → ∃𝑎𝑏((𝑎𝑉𝑏𝑉) ∧ (𝑎𝑏𝑌 = {𝑎, 𝑏}))))
26 r2ex 3043 . . . . . . . . . . . 12 (∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑌 = {𝑎, 𝑏}) ↔ ∃𝑎𝑏((𝑎𝑉𝑏𝑉) ∧ (𝑎𝑏𝑌 = {𝑎, 𝑏})))
2725, 26syl6ibr 241 . . . . . . . . . . 11 (𝑌 ∈ 𝒫 𝑉 → (∃𝑎𝑏(𝑎𝑏𝑌 = {𝑎, 𝑏}) → ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑌 = {𝑎, 𝑏})))
2827adantr 480 . . . . . . . . . 10 ((𝑌 ∈ 𝒫 𝑉 ∧ (#‘𝑌) = 2) → (∃𝑎𝑏(𝑎𝑏𝑌 = {𝑎, 𝑏}) → ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑌 = {𝑎, 𝑏})))
2913, 28mpd 15 . . . . . . . . 9 ((𝑌 ∈ 𝒫 𝑉 ∧ (#‘𝑌) = 2) → ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑌 = {𝑎, 𝑏}))
3012, 29sylbi 206 . . . . . . . 8 (𝑌 ∈ {𝑦 ∈ 𝒫 𝑉 ∣ (#‘𝑦) = 2} → ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑌 = {𝑎, 𝑏}))
319, 30syl6bi 242 . . . . . . 7 ((𝐸𝑥) = 𝑌 → ((𝐸𝑥) ∈ {𝑦 ∈ 𝒫 𝑉 ∣ (#‘𝑦) = 2} → ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑌 = {𝑎, 𝑏})))
328, 31syl5com 31 . . . . . 6 ((𝐸:dom 𝐸⟶{𝑦 ∈ 𝒫 𝑉 ∣ (#‘𝑦) = 2} ∧ 𝑥 ∈ dom 𝐸) → ((𝐸𝑥) = 𝑌 → ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑌 = {𝑎, 𝑏})))
3332ex 449 . . . . 5 (𝐸:dom 𝐸⟶{𝑦 ∈ 𝒫 𝑉 ∣ (#‘𝑦) = 2} → (𝑥 ∈ dom 𝐸 → ((𝐸𝑥) = 𝑌 → ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑌 = {𝑎, 𝑏}))))
346, 7, 333syl 18 . . . 4 (𝑉 USGrph 𝐸 → (𝑥 ∈ dom 𝐸 → ((𝐸𝑥) = 𝑌 → ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑌 = {𝑎, 𝑏}))))
3534rexlimdv 3012 . . 3 (𝑉 USGrph 𝐸 → (∃𝑥 ∈ dom 𝐸(𝐸𝑥) = 𝑌 → ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑌 = {𝑎, 𝑏})))
365, 35sylbid 229 . 2 (𝑉 USGrph 𝐸 → (𝑌 ∈ ran 𝐸 → ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑌 = {𝑎, 𝑏})))
3736imp 444 1 ((𝑉 USGrph 𝐸𝑌 ∈ ran 𝐸) → ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑌 = {𝑎, 𝑏}))
 Colors of variables: wff setvar class 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
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