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Theorem uspgrloopiedg 27313
Description: The set of edges in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 27045) is a singleton of a singleton. (Contributed by AV, 21-Feb-2021.)
Hypothesis
Ref Expression
uspgrloopvtx.g 𝐺 = ⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩
Assertion
Ref Expression
uspgrloopiedg ((𝑉𝑊𝐴𝑋) → (iEdg‘𝐺) = {⟨𝐴, {𝑁}⟩})

Proof of Theorem uspgrloopiedg
StepHypRef Expression
1 uspgrloopvtx.g . . 3 𝐺 = ⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩
21fveq2i 6664 . 2 (iEdg‘𝐺) = (iEdg‘⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩)
3 snex 5319 . . . 4 {⟨𝐴, {𝑁}⟩} ∈ V
43a1i 11 . . 3 (𝐴𝑋 → {⟨𝐴, {𝑁}⟩} ∈ V)
5 opiedgfv 26806 . . 3 ((𝑉𝑊 ∧ {⟨𝐴, {𝑁}⟩} ∈ V) → (iEdg‘⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩) = {⟨𝐴, {𝑁}⟩})
64, 5sylan2 595 . 2 ((𝑉𝑊𝐴𝑋) → (iEdg‘⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩) = {⟨𝐴, {𝑁}⟩})
72, 6syl5eq 2871 1 ((𝑉𝑊𝐴𝑋) → (iEdg‘𝐺) = {⟨𝐴, {𝑁}⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  Vcvv 3480  {csn 4550  cop 4556  cfv 6343  iEdgciedg 26796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-iota 6302  df-fun 6345  df-fv 6351  df-2nd 7685  df-iedg 26798
This theorem is referenced by:  uspgrloopnb0  27315  uspgrloopvd2  27316
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