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Theorem uspgrloopiedg 26648
Description: The set of edges in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 26364) 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 6336 . 2 (iEdg‘𝐺) = (iEdg‘⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩)
3 snex 5037 . . . 4 {⟨𝐴, {𝑁}⟩} ∈ V
43a1i 11 . . 3 (𝐴𝑋 → {⟨𝐴, {𝑁}⟩} ∈ V)
5 opiedgfv 26108 . . 3 ((𝑉𝑊 ∧ {⟨𝐴, {𝑁}⟩} ∈ V) → (iEdg‘⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩) = {⟨𝐴, {𝑁}⟩})
64, 5sylan2 580 . 2 ((𝑉𝑊𝐴𝑋) → (iEdg‘⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩) = {⟨𝐴, {𝑁}⟩})
72, 6syl5eq 2817 1 ((𝑉𝑊𝐴𝑋) → (iEdg‘𝐺) = {⟨𝐴, {𝑁}⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  Vcvv 3351  {csn 4317  cop 4323  cfv 6030  iEdgciedg 26096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-iota 5993  df-fun 6032  df-fv 6038  df-2nd 7320  df-iedg 26098
This theorem is referenced by:  uspgrloopnb0  26650  uspgrloopvd2  26651
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