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Theorem uspgrloopedg 26817
Description: The set of edges in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 26547) 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 6437 . . 3 (Edg‘𝐺) = (Edg‘⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩)
3 snex 5130 . . . . 5 {⟨𝐴, {𝑁}⟩} ∈ V
43a1i 11 . . . 4 (𝐴𝑋 → {⟨𝐴, {𝑁}⟩} ∈ V)
5 edgopval 26350 . . . 4 ((𝑉𝑊 ∧ {⟨𝐴, {𝑁}⟩} ∈ V) → (Edg‘⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩) = ran {⟨𝐴, {𝑁}⟩})
64, 5sylan2 588 . . 3 ((𝑉𝑊𝐴𝑋) → (Edg‘⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩) = ran {⟨𝐴, {𝑁}⟩})
72, 6syl5eq 2874 . 2 ((𝑉𝑊𝐴𝑋) → (Edg‘𝐺) = ran {⟨𝐴, {𝑁}⟩})
8 rnsnopg 5856 . . 3 (𝐴𝑋 → ran {⟨𝐴, {𝑁}⟩} = {{𝑁}})
98adantl 475 . 2 ((𝑉𝑊𝐴𝑋) → ran {⟨𝐴, {𝑁}⟩} = {{𝑁}})
107, 9eqtrd 2862 1 ((𝑉𝑊𝐴𝑋) → (Edg‘𝐺) = {{𝑁}})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1658  wcel 2166  Vcvv 3415  {csn 4398  cop 4404  ran crn 5344  cfv 6124  Edgcedg 26346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-br 4875  df-opab 4937  df-mpt 4954  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-iota 6087  df-fun 6126  df-fv 6132  df-2nd 7430  df-iedg 26298  df-edg 26347
This theorem is referenced by: (None)
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