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Theorem usgredg2ALT 29048
Description: Alternate proof of usgredg2 29047, not using umgredg2 28955. (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 2725 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
2 usgredg2.e . . . . 5 𝐸 = (iEdg‘𝐺)
31, 2usgrf 29010 . . . 4 (𝐺 ∈ USGraph → 𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
4 f1f 6787 . . . 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 7088 . 2 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → (𝐸𝑋) ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
7 fveq2 6891 . . . . 5 (𝑥 = (𝐸𝑋) → (♯‘𝑥) = (♯‘(𝐸𝑋)))
87eqeq1d 2727 . . . 4 (𝑥 = (𝐸𝑋) → ((♯‘𝑥) = 2 ↔ (♯‘(𝐸𝑋)) = 2))
98elrab 3675 . . 3 ((𝐸𝑋) ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2} ↔ ((𝐸𝑋) ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ (♯‘(𝐸𝑋)) = 2))
109simprbi 495 . 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 394   = wceq 1533  wcel 2098  {crab 3419  cdif 3937  c0 4318  𝒫 cpw 4598  {csn 4624  dom cdm 5672  wf 6538  1-1wf1 6539  cfv 6542  2c2 12295  chash 14319  Vtxcvtx 28851  iEdgciedg 28852  USGraphcusgr 29004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3770  df-dif 3943  df-un 3945  df-ss 3957  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fv 6550  df-usgr 29006
This theorem is referenced by: (None)
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