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Theorem s1val 13943
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 13941 . 2 ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
2 fvi 6722 . . . 4 (𝐴𝑉 → ( I ‘𝐴) = 𝐴)
32opeq2d 4785 . . 3 (𝐴𝑉 → ⟨0, ( I ‘𝐴)⟩ = ⟨0, 𝐴⟩)
43sneqd 4551 . 2 (𝐴𝑉 → {⟨0, ( I ‘𝐴)⟩} = {⟨0, 𝐴⟩})
51, 4syl5eq 2869 1 (𝐴𝑉 → ⟨“𝐴”⟩ = {⟨0, 𝐴⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2114  {csn 4539  cop 4545   I cid 5436  cfv 6334  0cc0 10526  ⟨“cs1 13940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-iota 6293  df-fun 6336  df-fv 6342  df-s1 13941
This theorem is referenced by:  s1rn  13944  s1cl  13947  s1dmALT  13954  s1fv  13955  s111  13960  repsw1  14136  s1co  14186  s2prop  14260  ofs1  14321  gsumws1  17993  uspgr1ewop  27036  usgr2v1e2w  27040  0wlkons1  27904  s1f1  30629  cshw1s2  30644  ofcs1  31888  signstf0  31912
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