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Theorem usgredg2ALT 26305
Description: Alternate proof of usgredg2 26304, not using umgredg2 26215. (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 2760 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
2 usgredg2.e . . . . 5 𝐸 = (iEdg‘𝐺)
31, 2usgrf 26270 . . . 4 (𝐺 ∈ USGraph → 𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
4 f1f 6262 . . . 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})
65ffvelrnda 6523 . 2 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → (𝐸𝑋) ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
7 fveq2 6353 . . . . 5 (𝑥 = (𝐸𝑋) → (♯‘𝑥) = (♯‘(𝐸𝑋)))
87eqeq1d 2762 . . . 4 (𝑥 = (𝐸𝑋) → ((♯‘𝑥) = 2 ↔ (♯‘(𝐸𝑋)) = 2))
98elrab 3504 . . 3 ((𝐸𝑋) ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2} ↔ ((𝐸𝑋) ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ (♯‘(𝐸𝑋)) = 2))
109simprbi 483 . 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 383   = wceq 1632  wcel 2139  {crab 3054  cdif 3712  c0 4058  𝒫 cpw 4302  {csn 4321  dom cdm 5266  wf 6045  1-1wf1 6046  cfv 6049  2c2 11282  chash 13331  Vtxcvtx 26094  iEdgciedg 26095  USGraphcusgr 26264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fv 6057  df-usgr 26266
This theorem is referenced by: (None)
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