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Theorem usgraedg2 25904
 Description: The value of the "edge function" of a graph is a set containing two elements (the vertices the corresponding edge is connecting), analogous to umgrale 25850. (Contributed by Alexander van der Vekens, 11-Aug-2017.)
Assertion
Ref Expression
usgraedg2 ((𝑉 USGrph 𝐸𝑋 ∈ dom 𝐸) → (#‘(𝐸𝑋)) = 2)

Proof of Theorem usgraedg2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 usgraf 25875 . . . 4 (𝑉 USGrph 𝐸𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2})
2 f1f 6014 . . . 4 (𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2} → 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2})
31, 2syl 17 . . 3 (𝑉 USGrph 𝐸𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2})
43ffvelrnda 6267 . 2 ((𝑉 USGrph 𝐸𝑋 ∈ dom 𝐸) → (𝐸𝑋) ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2})
5 fveq2 6103 . . . . 5 (𝑥 = (𝐸𝑋) → (#‘𝑥) = (#‘(𝐸𝑋)))
65eqeq1d 2612 . . . 4 (𝑥 = (𝐸𝑋) → ((#‘𝑥) = 2 ↔ (#‘(𝐸𝑋)) = 2))
76elrab 3331 . . 3 ((𝐸𝑋) ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2} ↔ ((𝐸𝑋) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (#‘(𝐸𝑋)) = 2))
87simprbi 479 . 2 ((𝐸𝑋) ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2} → (#‘(𝐸𝑋)) = 2)
94, 8syl 17 1 ((𝑉 USGrph 𝐸𝑋 ∈ dom 𝐸) → (#‘(𝐸𝑋)) = 2)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900   ∖ cdif 3537  ∅c0 3874  𝒫 cpw 4108  {csn 4125   class class class wbr 4583  dom cdm 5038  ⟶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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  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-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fv 5812  df-usgra 25862 This theorem is referenced by:  usgraedgprv  25905  usgranloopv  25907
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