Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  s1eq Structured version   Visualization version   GIF version

Theorem s1eq 13570
 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 6352 . . . 4 (𝐴 = 𝐵 → ( I ‘𝐴) = ( I ‘𝐵))
21opeq2d 4560 . . 3 (𝐴 = 𝐵 → ⟨0, ( I ‘𝐴)⟩ = ⟨0, ( I ‘𝐵)⟩)
32sneqd 4333 . 2 (𝐴 = 𝐵 → {⟨0, ( I ‘𝐴)⟩} = {⟨0, ( I ‘𝐵)⟩})
4 df-s1 13488 . 2 ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
5 df-s1 13488 . 2 ⟨“𝐵”⟩ = {⟨0, ( I ‘𝐵)⟩}
63, 4, 53eqtr4g 2819 1 (𝐴 = 𝐵 → ⟨“𝐴”⟩ = ⟨“𝐵”⟩)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1632  {csn 4321  ⟨cop 4327   I cid 5173  ‘cfv 6049  0cc0 10128  ⟨“cs1 13480 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-iota 6012  df-fv 6057  df-s1 13488 This theorem is referenced by:  s1eqd  13571  wrdl1exs1  13584  wrdl1s1  13585  wrdind  13676  wrd2ind  13677  ccats1swrdeqrex  13678  reuccats1lem  13679  reuccats1  13680  revs1  13714  vrmdval  17595  frgpup3lem  18390  vdegp1ci  26644  clwwlknonwwlknonb  27254  clwwlknonwwlknonbOLD  27255  mrsubcv  31714  mrsubrn  31717  elmrsubrn  31724  mrsubvrs  31726  mvhval  31738  ccats1pfxeqrex  41932  reuccatpfxs1lem  41943  reuccatpfxs1  41944
 Copyright terms: Public domain W3C validator