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Theorem s1val 14572
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 14570 . 2 ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
2 fvi 6968 . . . 4 (𝐴𝑉 → ( I ‘𝐴) = 𝐴)
32opeq2d 4876 . . 3 (𝐴𝑉 → ⟨0, ( I ‘𝐴)⟩ = ⟨0, 𝐴⟩)
43sneqd 4636 . 2 (𝐴𝑉 → {⟨0, ( I ‘𝐴)⟩} = {⟨0, 𝐴⟩})
51, 4eqtrid 2779 1 (𝐴𝑉 → ⟨“𝐴”⟩ = {⟨0, 𝐴⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  {csn 4624  cop 4630   I cid 5569  cfv 6542  0cc0 11130  ⟨“cs1 14569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fv 6550  df-s1 14570
This theorem is referenced by:  s1rn  14573  s1cl  14576  s1dmALT  14583  s1fv  14584  s111  14589  repsw1  14757  s1co  14808  s2prop  14882  ofs1  14941  gsumws1  18781  uspgr1ewop  29048  usgr2v1e2w  29052  0wlkons1  29918  s1f1  32648  cshw1s2  32663  ofcs1  34112  signstf0  34136
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