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Theorem uspgrloopedg 29452
Description: The set of edges in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 29182) is a singleton of a singleton. (Contributed by AV, 17-Dec-2020.)
Hypothesis
Ref Expression
uspgrloopvtx.g 𝐺 = ⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩
Assertion
Ref Expression
uspgrloopedg ((𝑉𝑊𝐴𝑋) → (Edg‘𝐺) = {{𝑁}})

Proof of Theorem uspgrloopedg
StepHypRef Expression
1 uspgrloopvtx.g . . . 4 𝐺 = ⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩
21fveq2i 6896 . . 3 (Edg‘𝐺) = (Edg‘⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩)
3 snex 5429 . . . . 5 {⟨𝐴, {𝑁}⟩} ∈ V
43a1i 11 . . . 4 (𝐴𝑋 → {⟨𝐴, {𝑁}⟩} ∈ V)
5 edgopval 28984 . . . 4 ((𝑉𝑊 ∧ {⟨𝐴, {𝑁}⟩} ∈ V) → (Edg‘⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩) = ran {⟨𝐴, {𝑁}⟩})
64, 5sylan2 591 . . 3 ((𝑉𝑊𝐴𝑋) → (Edg‘⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩) = ran {⟨𝐴, {𝑁}⟩})
72, 6eqtrid 2778 . 2 ((𝑉𝑊𝐴𝑋) → (Edg‘𝐺) = ran {⟨𝐴, {𝑁}⟩})
8 rnsnopg 6224 . . 3 (𝐴𝑋 → ran {⟨𝐴, {𝑁}⟩} = {{𝑁}})
98adantl 480 . 2 ((𝑉𝑊𝐴𝑋) → ran {⟨𝐴, {𝑁}⟩} = {{𝑁}})
107, 9eqtrd 2766 1 ((𝑉𝑊𝐴𝑋) → (Edg‘𝐺) = {{𝑁}})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  Vcvv 3462  {csn 4623  cop 4629  ran crn 5675  cfv 6546  Edgcedg 28980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-iota 6498  df-fun 6548  df-fv 6554  df-2nd 7996  df-iedg 28932  df-edg 28981
This theorem is referenced by: (None)
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