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Theorem iedgvalsnop 25717
 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.)
Hypotheses
Ref Expression
vtxvalsnop.b 𝐵 ∈ V
vtxvalsnop.g 𝐺 = {⟨𝐵, 𝐵⟩}
Assertion
Ref Expression
iedgvalsnop (iEdg‘𝐺) = {𝐵}

Proof of Theorem iedgvalsnop
StepHypRef Expression
1 eqid 2610 . . 3 𝐵 = 𝐵
2 vtxvalsnop.b . . . 4 𝐵 ∈ V
3 vtxvalsnop.g . . . 4 𝐺 = {⟨𝐵, 𝐵⟩}
42, 2, 3funsneqopsn 6322 . . 3 (𝐵 = 𝐵𝐺 = ⟨{𝐵}, {𝐵}⟩)
51, 4ax-mp 5 . 2 𝐺 = ⟨{𝐵}, {𝐵}⟩
6 fveq2 6103 . . 3 (𝐺 = ⟨{𝐵}, {𝐵}⟩ → (iEdg‘𝐺) = (iEdg‘⟨{𝐵}, {𝐵}⟩))
7 opex 4859 . . . 4 ⟨{𝐵}, {𝐵}⟩ ∈ V
8 iedgval 25678 . . . . 5 (⟨{𝐵}, {𝐵}⟩ ∈ V → (iEdg‘⟨{𝐵}, {𝐵}⟩) = if(⟨{𝐵}, {𝐵}⟩ ∈ (V × V), (2nd ‘⟨{𝐵}, {𝐵}⟩), (.ef‘⟨{𝐵}, {𝐵}⟩)))
9 snex 4835 . . . . . . . 8 {𝐵} ∈ V
109, 9opelvv 5088 . . . . . . 7 ⟨{𝐵}, {𝐵}⟩ ∈ (V × V)
1110iftruei 4043 . . . . . 6 if(⟨{𝐵}, {𝐵}⟩ ∈ (V × V), (2nd ‘⟨{𝐵}, {𝐵}⟩), (.ef‘⟨{𝐵}, {𝐵}⟩)) = (2nd ‘⟨{𝐵}, {𝐵}⟩)
129, 9op2nd 7068 . . . . . 6 (2nd ‘⟨{𝐵}, {𝐵}⟩) = {𝐵}
1311, 12eqtri 2632 . . . . 5 if(⟨{𝐵}, {𝐵}⟩ ∈ (V × V), (2nd ‘⟨{𝐵}, {𝐵}⟩), (.ef‘⟨{𝐵}, {𝐵}⟩)) = {𝐵}
148, 13syl6eq 2660 . . . 4 (⟨{𝐵}, {𝐵}⟩ ∈ V → (iEdg‘⟨{𝐵}, {𝐵}⟩) = {𝐵})
157, 14ax-mp 5 . . 3 (iEdg‘⟨{𝐵}, {𝐵}⟩) = {𝐵}
166, 15syl6eq 2660 . 2 (𝐺 = ⟨{𝐵}, {𝐵}⟩ → (iEdg‘𝐺) = {𝐵})
175, 16ax-mp 5 1 (iEdg‘𝐺) = {𝐵}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ifcif 4036  {csn 4125  ⟨cop 4131   × cxp 5036  ‘cfv 5804  2nd c2nd 7058  .efcedgf 25667  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:  iedgval3sn  25719
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