Theorem iedgval3sn 25719

Description: Degenerated case 3 for edges: The set of indexed edges of a singleton containing a singleton containing a singleton is the innermost singleton. (Contributed by AV, 24-Sep-2020.)

Hypothesis

Ref	Expression
vtxval3sn.a	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
iedgval3sn	⊢ (iEdg‘{{{𝐴}}}) = {𝐴}

Proof of Theorem iedgval3sn

Step	Hyp	Ref	Expression
1		vtxval3sn.a	. 2 ⊢ 𝐴 ∈ V
2	1	opid 4359	. . . 4 ⊢ ⟨𝐴, 𝐴⟩ = {{𝐴}}
3	2	eqcomi 2619	. . 3 ⊢ {{𝐴}} = ⟨𝐴, 𝐴⟩
4	3	sneqi 4136	. 2 ⊢ {{{𝐴}}} = {⟨𝐴, 𝐴⟩}
5	1, 4	iedgvalsnop 25717	1 ⊢ (iEdg‘{{{𝐴}}}) = {𝐴}

Syntax hints:   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {csn 4125  ⟨cop 4131  ‘cfv 5804  iEdgciedg 25674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-2nd 7060  df-iedg 25676

This theorem is referenced by: (None)