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Theorem uspgrloopedg 29371
Description: The set of edges in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 29101) 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 6893 . . 3 (Edg‘𝐺) = (Edg‘⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩)
3 snex 5428 . . . . 5 {⟨𝐴, {𝑁}⟩} ∈ V
43a1i 11 . . . 4 (𝐴𝑋 → {⟨𝐴, {𝑁}⟩} ∈ V)
5 edgopval 28903 . . . 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 2777 . 2 ((𝑉𝑊𝐴𝑋) → (Edg‘𝐺) = ran {⟨𝐴, {𝑁}⟩})
8 rnsnopg 6221 . . 3 (𝐴𝑋 → ran {⟨𝐴, {𝑁}⟩} = {{𝑁}})
98adantl 480 . 2 ((𝑉𝑊𝐴𝑋) → ran {⟨𝐴, {𝑁}⟩} = {{𝑁}})
107, 9eqtrd 2765 1 ((𝑉𝑊𝐴𝑋) → (Edg‘𝐺) = {{𝑁}})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  Vcvv 3463  {csn 4625  cop 4631  ran crn 5674  cfv 6543  Edgcedg 28899
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-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pr 5424  ax-un 7735
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-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-iota 6495  df-fun 6545  df-fv 6551  df-2nd 7988  df-iedg 28851  df-edg 28900
This theorem is referenced by: (None)
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