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Theorem s1val 13414
Description: Value of a singleton word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
Assertion
Ref Expression
s1val (𝐴𝑉 → ⟨“𝐴”⟩ = {⟨0, 𝐴⟩})

Proof of Theorem s1val
StepHypRef Expression
1 df-s1 13334 . 2 ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
2 fvi 6294 . . . 4 (𝐴𝑉 → ( I ‘𝐴) = 𝐴)
32opeq2d 4440 . . 3 (𝐴𝑉 → ⟨0, ( I ‘𝐴)⟩ = ⟨0, 𝐴⟩)
43sneqd 4222 . 2 (𝐴𝑉 → {⟨0, ( I ‘𝐴)⟩} = {⟨0, 𝐴⟩})
51, 4syl5eq 2697 1 (𝐴𝑉 → ⟨“𝐴”⟩ = {⟨0, 𝐴⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  {csn 4210  cop 4216   I cid 5052  cfv 5926  0cc0 9974  ⟨“cs1 13326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-s1 13334
This theorem is referenced by:  s1rn  13415  s1cl  13418  s1dmALT  13426  s1fv  13427  s111  13432  repsw1  13576  s1co  13625  s2prop  13698  ofs1  13755  gsumws1  17423  uspgr1ewop  26185  usgr2v1e2w  26189  0wlkons1  27099  ofcs1  30749  signstf0  30773
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