Theorem iedgvalsnop 26827

Description: Degenerated case 2 for edges: The set of indexed edges of a singleton containing an ordered pair with equal components is the singleton containing the component. (Contributed by AV, 24-Sep-2020.) (Proof shortened by AV, 15-Jul-2022.) (Avoid depending on this detail.)

Hypotheses

Ref	Expression
vtxvalsnop.b	⊢ 𝐵 ∈ V
vtxvalsnop.g	⊢ 𝐺 = {⟨𝐵, 𝐵⟩}

Assertion

Ref	Expression
iedgvalsnop	⊢ (iEdg‘𝐺) = {𝐵}

Proof of Theorem iedgvalsnop

Step	Hyp	Ref	Expression
1		vtxvalsnop.g	. . 3 ⊢ 𝐺 = {⟨𝐵, 𝐵⟩}
2	1	fveq2i 6673	. 2 ⊢ (iEdg‘𝐺) = (iEdg‘{⟨𝐵, 𝐵⟩})
3		vtxvalsnop.b	. . . 4 ⊢ 𝐵 ∈ V
4	3	snopeqopsnid 5399	. . 3 ⊢ {⟨𝐵, 𝐵⟩} = ⟨{𝐵}, {𝐵}⟩
5	4	fveq2i 6673	. 2 ⊢ (iEdg‘{⟨𝐵, 𝐵⟩}) = (iEdg‘⟨{𝐵}, {𝐵}⟩)
6		snex 5332	. . 3 ⊢ {𝐵} ∈ V
7	6, 6	opiedgfvi 26795	. 2 ⊢ (iEdg‘⟨{𝐵}, {𝐵}⟩) = {𝐵}
8	2, 5, 7	3eqtri 2848	1 ⊢ (iEdg‘𝐺) = {𝐵}

Syntax hints:   = wceq 1537   ∈ wcel 2114  Vcvv 3494  {csn 4567  ⟨cop 4573  ‘cfv 6355  iEdgciedg 26782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-iota 6314  df-fun 6357  df-fv 6363  df-2nd 7690  df-iedg 26784

This theorem is referenced by:  iedgval3sn  26829