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Theorem s1val 13946
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 13944 . 2 ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
2 fvi 6734 . . . 4 (𝐴𝑉 → ( I ‘𝐴) = 𝐴)
32opeq2d 4803 . . 3 (𝐴𝑉 → ⟨0, ( I ‘𝐴)⟩ = ⟨0, 𝐴⟩)
43sneqd 4572 . 2 (𝐴𝑉 → {⟨0, ( I ‘𝐴)⟩} = {⟨0, 𝐴⟩})
51, 4syl5eq 2868 1 (𝐴𝑉 → ⟨“𝐴”⟩ = {⟨0, 𝐴⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2110  {csn 4560  cop 4566   I cid 5453  cfv 6349  0cc0 10531  ⟨“cs1 13943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-iota 6308  df-fun 6351  df-fv 6357  df-s1 13944
This theorem is referenced by:  s1rn  13947  s1cl  13950  s1dmALT  13957  s1fv  13958  s111  13963  repsw1  14139  s1co  14189  s2prop  14263  ofs1  14324  gsumws1  17996  uspgr1ewop  27024  usgr2v1e2w  27028  0wlkons1  27894  s1f1  30614  cshw1s2  30629  ofcs1  31809  signstf0  31833
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