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Theorem usgredg2ALT 29225
Description: Alternate proof of usgredg2 29224, not using umgredg2 29132. (Contributed by Alexander van der Vekens, 11-Aug-2017.) (Revised by AV, 16-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
usgredg2.e 𝐸 = (iEdg‘𝐺)
Assertion
Ref Expression
usgredg2ALT ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → (♯‘(𝐸𝑋)) = 2)

Proof of Theorem usgredg2ALT
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqid 2735 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
2 usgredg2.e . . . . 5 𝐸 = (iEdg‘𝐺)
31, 2usgrf 29187 . . . 4 (𝐺 ∈ USGraph → 𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
4 f1f 6805 . . . 4 (𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2} → 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
53, 4syl 17 . . 3 (𝐺 ∈ USGraph → 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
65ffvelcdmda 7104 . 2 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → (𝐸𝑋) ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
7 fveq2 6907 . . . . 5 (𝑥 = (𝐸𝑋) → (♯‘𝑥) = (♯‘(𝐸𝑋)))
87eqeq1d 2737 . . . 4 (𝑥 = (𝐸𝑋) → ((♯‘𝑥) = 2 ↔ (♯‘(𝐸𝑋)) = 2))
98elrab 3695 . . 3 ((𝐸𝑋) ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2} ↔ ((𝐸𝑋) ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ (♯‘(𝐸𝑋)) = 2))
109simprbi 496 . 2 ((𝐸𝑋) ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2} → (♯‘(𝐸𝑋)) = 2)
116, 10syl 17 1 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → (♯‘(𝐸𝑋)) = 2)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  {crab 3433  cdif 3960  c0 4339  𝒫 cpw 4605  {csn 4631  dom cdm 5689  wf 6559  1-1wf1 6560  cfv 6563  2c2 12319  chash 14366  Vtxcvtx 29028  iEdgciedg 29029  USGraphcusgr 29181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fv 6571  df-usgr 29183
This theorem is referenced by: (None)
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