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Theorem edgss 25881
 Description: The set of edges of an undirected simple graph without loops is a subset of the set of unordered pairs of vertices. (Contributed by AV, 1-Jan-2020.)
Assertion
Ref Expression
edgss (𝐺 ∈ USGrph → (Edges‘𝐺) ⊆ {𝑥 ∈ 𝒫 (1st𝐺) ∣ (#‘𝑥) = 2})
Distinct variable group:   𝑥,𝐺

Proof of Theorem edgss
Dummy variables 𝑒 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 edgval 25868 . 2 (𝐺 ∈ USGrph → (Edges‘𝐺) = ran (2nd𝐺))
2 usgraop 25879 . . 3 (𝐺 ∈ USGrph → ∃𝑣𝑒(𝐺 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:dom 𝑒1-1→{𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2}))
3 f1f 6014 . . . . . . 7 (𝑒:dom 𝑒1-1→{𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2} → 𝑒:dom 𝑒⟶{𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2})
4 frn 5966 . . . . . . 7 (𝑒:dom 𝑒⟶{𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2} → ran 𝑒 ⊆ {𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2})
53, 4syl 17 . . . . . 6 (𝑒:dom 𝑒1-1→{𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2} → ran 𝑒 ⊆ {𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2})
65adantl 481 . . . . 5 ((𝐺 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:dom 𝑒1-1→{𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2}) → ran 𝑒 ⊆ {𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2})
7 vex 3176 . . . . . . . . 9 𝑣 ∈ V
8 vex 3176 . . . . . . . . 9 𝑒 ∈ V
97, 8op2ndd 7070 . . . . . . . 8 (𝐺 = ⟨𝑣, 𝑒⟩ → (2nd𝐺) = 𝑒)
109rneqd 5274 . . . . . . 7 (𝐺 = ⟨𝑣, 𝑒⟩ → ran (2nd𝐺) = ran 𝑒)
117, 8op1std 7069 . . . . . . . . 9 (𝐺 = ⟨𝑣, 𝑒⟩ → (1st𝐺) = 𝑣)
1211pweqd 4113 . . . . . . . 8 (𝐺 = ⟨𝑣, 𝑒⟩ → 𝒫 (1st𝐺) = 𝒫 𝑣)
13 rabeq 3166 . . . . . . . 8 (𝒫 (1st𝐺) = 𝒫 𝑣 → {𝑥 ∈ 𝒫 (1st𝐺) ∣ (#‘𝑥) = 2} = {𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2})
1412, 13syl 17 . . . . . . 7 (𝐺 = ⟨𝑣, 𝑒⟩ → {𝑥 ∈ 𝒫 (1st𝐺) ∣ (#‘𝑥) = 2} = {𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2})
1510, 14sseq12d 3597 . . . . . 6 (𝐺 = ⟨𝑣, 𝑒⟩ → (ran (2nd𝐺) ⊆ {𝑥 ∈ 𝒫 (1st𝐺) ∣ (#‘𝑥) = 2} ↔ ran 𝑒 ⊆ {𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2}))
1615adantr 480 . . . . 5 ((𝐺 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:dom 𝑒1-1→{𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2}) → (ran (2nd𝐺) ⊆ {𝑥 ∈ 𝒫 (1st𝐺) ∣ (#‘𝑥) = 2} ↔ ran 𝑒 ⊆ {𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2}))
176, 16mpbird 246 . . . 4 ((𝐺 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:dom 𝑒1-1→{𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2}) → ran (2nd𝐺) ⊆ {𝑥 ∈ 𝒫 (1st𝐺) ∣ (#‘𝑥) = 2})
1817exlimivv 1847 . . 3 (∃𝑣𝑒(𝐺 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:dom 𝑒1-1→{𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2}) → ran (2nd𝐺) ⊆ {𝑥 ∈ 𝒫 (1st𝐺) ∣ (#‘𝑥) = 2})
192, 18syl 17 . 2 (𝐺 ∈ USGrph → ran (2nd𝐺) ⊆ {𝑥 ∈ 𝒫 (1st𝐺) ∣ (#‘𝑥) = 2})
201, 19eqsstrd 3602 1 (𝐺 ∈ USGrph → (Edges‘𝐺) ⊆ {𝑥 ∈ 𝒫 (1st𝐺) ∣ (#‘𝑥) = 2})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {crab 2900   ⊆ wss 3540  𝒫 cpw 4108  ⟨cop 4131  dom cdm 5038  ran crn 5039  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804  1st c1st 7057  2nd c2nd 7058  2c2 10947  #chash 12979   USGrph cusg 25859  Edgescedg 25860 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  df-edg 25865 This theorem is referenced by:  edg  25882
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