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Theorem s1eq 13763
 Description: Equality theorem for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
Assertion
Ref Expression
s1eq (𝐴 = 𝐵 → ⟨“𝐴”⟩ = ⟨“𝐵”⟩)

Proof of Theorem s1eq
StepHypRef Expression
1 fveq2 6499 . . . 4 (𝐴 = 𝐵 → ( I ‘𝐴) = ( I ‘𝐵))
21opeq2d 4684 . . 3 (𝐴 = 𝐵 → ⟨0, ( I ‘𝐴)⟩ = ⟨0, ( I ‘𝐵)⟩)
32sneqd 4453 . 2 (𝐴 = 𝐵 → {⟨0, ( I ‘𝐴)⟩} = {⟨0, ( I ‘𝐵)⟩})
4 df-s1 13759 . 2 ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
5 df-s1 13759 . 2 ⟨“𝐵”⟩ = {⟨0, ( I ‘𝐵)⟩}
63, 4, 53eqtr4g 2839 1 (𝐴 = 𝐵 → ⟨“𝐴”⟩ = ⟨“𝐵”⟩)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1507  {csn 4441  ⟨cop 4447   I cid 5311  ‘cfv 6188  0cc0 10335  ⟨“cs1 13758 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2750 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-rex 3094  df-rab 3097  df-v 3417  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-iota 6152  df-fv 6196  df-s1 13759 This theorem is referenced by:  s1eqd  13764  wrdl1exs1  13776  wrdl1s1  13777  ccats1pfxeqrex  13906  wrdind  13915  wrdindOLD  13916  wrd2ind  13917  wrd2indOLD  13918  ccats1swrdeqrexOLD  13919  reuccats1lemOLD  13920  reuccats1OLD  13921  reuccatpfxs1lem  13955  reuccatpfxs1  13956  revs1  13984  vrmdval  17863  frgpup3lem  18663  vdegp1ci  27023  clwwlknonwwlknonb  27634  mrsubcv  32283  mrsubrn  32286  elmrsubrn  32293  mrsubvrs  32295  mvhval  32307
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