Theorem iedgvalsnop 29243

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 6870	. 2 ⊢ (iEdg‘𝐺) = (iEdg‘{⟨𝐵, 𝐵⟩})
3		vtxvalsnop.b	. . . 4 ⊢ 𝐵 ∈ V
4	3	snopeqopsnid 5478	. . 3 ⊢ {⟨𝐵, 𝐵⟩} = ⟨{𝐵}, {𝐵}⟩
5	4	fveq2i 6870	. 2 ⊢ (iEdg‘{⟨𝐵, 𝐵⟩}) = (iEdg‘⟨{𝐵}, {𝐵}⟩)
6		snex 5396	. . 3 ⊢ {𝐵} ∈ V
7	6, 6	opiedgfvi 29211	. 2 ⊢ (iEdg‘⟨{𝐵}, {𝐵}⟩) = {𝐵}
8	2, 5, 7	3eqtri 2789	1 ⊢ (iEdg‘𝐺) = {𝐵}

Syntax hints:   = wceq 1560   ∈ wcel 2142  Vcvv 3454  {csn 4582  ⟨cop 4588  ‘cfv 6521  iEdgciedg 29198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-iota 6477  df-fun 6523  df-fv 6529  df-2nd 7971  df-iedg 29200

This theorem is referenced by:  iedgval3sn  29245