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Theorem usgredg2ALT 29210
Description: Alternate proof of usgredg2 29209, not using umgredg2 29117. (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 2737 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
2 usgredg2.e . . . . 5 𝐸 = (iEdg‘𝐺)
31, 2usgrf 29172 . . . 4 (𝐺 ∈ USGraph → 𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
4 f1f 6804 . . . 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 6906 . . . . 5 (𝑥 = (𝐸𝑋) → (♯‘𝑥) = (♯‘(𝐸𝑋)))
87eqeq1d 2739 . . . 4 (𝑥 = (𝐸𝑋) → ((♯‘𝑥) = 2 ↔ (♯‘(𝐸𝑋)) = 2))
98elrab 3692 . . 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 1540  wcel 2108  {crab 3436  cdif 3948  c0 4333  𝒫 cpw 4600  {csn 4626  dom cdm 5685  wf 6557  1-1wf1 6558  cfv 6561  2c2 12321  chash 14369  Vtxcvtx 29013  iEdgciedg 29014  USGraphcusgr 29166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fv 6569  df-usgr 29168
This theorem is referenced by: (None)
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