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Theorem usgraop 25879
 Description: An undirected simple graph without loops represented as ordered pair with a one-to-one edge function. (Contributed by AV, 1-Jan-2020.)
Assertion
Ref Expression
usgraop (𝐺 ∈ USGrph → ∃𝑣𝑒(𝐺 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:dom 𝑒1-1→{𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2}))
Distinct variable group:   𝑒,𝐺,𝑣,𝑥

Proof of Theorem usgraop
StepHypRef Expression
1 df-usgra 25862 . . . 4 USGrph = {⟨𝑣, 𝑒⟩ ∣ 𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2}}
21eleq2i 2680 . . 3 (𝐺 ∈ USGrph ↔ 𝐺 ∈ {⟨𝑣, 𝑒⟩ ∣ 𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2}})
3 elopab 4908 . . 3 (𝐺 ∈ {⟨𝑣, 𝑒⟩ ∣ 𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2}} ↔ ∃𝑣𝑒(𝐺 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2}))
42, 3bitri 263 . 2 (𝐺 ∈ USGrph ↔ ∃𝑣𝑒(𝐺 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2}))
5 vex 3176 . . . . . . . . . . . . . . 15 𝑥 ∈ V
6 hasheq0 13015 . . . . . . . . . . . . . . 15 (𝑥 ∈ V → ((#‘𝑥) = 0 ↔ 𝑥 = ∅))
75, 6ax-mp 5 . . . . . . . . . . . . . 14 ((#‘𝑥) = 0 ↔ 𝑥 = ∅)
8 2ne0 10990 . . . . . . . . . . . . . . . . . 18 2 ≠ 0
98a1i 11 . . . . . . . . . . . . . . . . 17 ((#‘𝑥) = 0 → 2 ≠ 0)
10 id 22 . . . . . . . . . . . . . . . . 17 ((#‘𝑥) = 0 → (#‘𝑥) = 0)
119, 10neeqtrrd 2856 . . . . . . . . . . . . . . . 16 ((#‘𝑥) = 0 → 2 ≠ (#‘𝑥))
1211necomd 2837 . . . . . . . . . . . . . . 15 ((#‘𝑥) = 0 → (#‘𝑥) ≠ 2)
1312neneqd 2787 . . . . . . . . . . . . . 14 ((#‘𝑥) = 0 → ¬ (#‘𝑥) = 2)
147, 13sylbir 224 . . . . . . . . . . . . 13 (𝑥 = ∅ → ¬ (#‘𝑥) = 2)
1514con2i 133 . . . . . . . . . . . 12 ((#‘𝑥) = 2 → ¬ 𝑥 = ∅)
16 velsn 4141 . . . . . . . . . . . 12 (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
1715, 16sylnibr 318 . . . . . . . . . . 11 ((#‘𝑥) = 2 → ¬ 𝑥 ∈ {∅})
1817biantrud 527 . . . . . . . . . 10 ((#‘𝑥) = 2 → (𝑥 ∈ 𝒫 𝑣 ↔ (𝑥 ∈ 𝒫 𝑣 ∧ ¬ 𝑥 ∈ {∅})))
19 eldif 3550 . . . . . . . . . 10 (𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ↔ (𝑥 ∈ 𝒫 𝑣 ∧ ¬ 𝑥 ∈ {∅}))
2018, 19syl6rbbr 278 . . . . . . . . 9 ((#‘𝑥) = 2 → (𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ↔ 𝑥 ∈ 𝒫 𝑣))
2120pm5.32ri 668 . . . . . . . 8 ((𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∧ (#‘𝑥) = 2) ↔ (𝑥 ∈ 𝒫 𝑣 ∧ (#‘𝑥) = 2))
2221abbii 2726 . . . . . . 7 {𝑥 ∣ (𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∧ (#‘𝑥) = 2)} = {𝑥 ∣ (𝑥 ∈ 𝒫 𝑣 ∧ (#‘𝑥) = 2)}
23 df-rab 2905 . . . . . . 7 {𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2} = {𝑥 ∣ (𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∧ (#‘𝑥) = 2)}
24 df-rab 2905 . . . . . . 7 {𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2} = {𝑥 ∣ (𝑥 ∈ 𝒫 𝑣 ∧ (#‘𝑥) = 2)}
2522, 23, 243eqtr4i 2642 . . . . . 6 {𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2} = {𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2}
26 f1eq3 6011 . . . . . 6 ({𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2} = {𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2} → (𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2} ↔ 𝑒:dom 𝑒1-1→{𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2}))
2725, 26ax-mp 5 . . . . 5 (𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2} ↔ 𝑒:dom 𝑒1-1→{𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2})
2827biimpi 205 . . . 4 (𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2} → 𝑒:dom 𝑒1-1→{𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2})
2928anim2i 591 . . 3 ((𝐺 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2}) → (𝐺 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:dom 𝑒1-1→{𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2}))
30292eximi 1753 . 2 (∃𝑣𝑒(𝐺 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2}) → ∃𝑣𝑒(𝐺 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:dom 𝑒1-1→{𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2}))
314, 30sylbi 206 1 (𝐺 ∈ USGrph → ∃𝑣𝑒(𝐺 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:dom 𝑒1-1→{𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2}))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596   ≠ wne 2780  {crab 2900  Vcvv 3173   ∖ cdif 3537  ∅c0 3874  𝒫 cpw 4108  {csn 4125  ⟨cop 4131  {copab 4642  dom cdm 5038  –1-1→wf1 5801  ‘cfv 5804  0cc0 9815  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-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-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-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  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:  usgrac  25880  edgss  25881
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