Theorem s1eq 14565

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

Step	Hyp	Ref	Expression
1		fveq2 6858	. . . 4 ⊢ (𝐴 = 𝐵 → ( I ‘𝐴) = ( I ‘𝐵))
2	1	opeq2d 4844	. . 3 ⊢ (𝐴 = 𝐵 → ⟨0, ( I ‘𝐴)⟩ = ⟨0, ( I ‘𝐵)⟩)
3	2	sneqd 4601	. 2 ⊢ (𝐴 = 𝐵 → {⟨0, ( I ‘𝐴)⟩} = {⟨0, ( I ‘𝐵)⟩})
4		df-s1 14561	. 2 ⊢ ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
5		df-s1 14561	. 2 ⊢ ⟨“𝐵”⟩ = {⟨0, ( I ‘𝐵)⟩}
6	3, 4, 5	3eqtr4g 2789	1 ⊢ (𝐴 = 𝐵 → ⟨“𝐴”⟩ = ⟨“𝐵”⟩)

Syntax hints:   → wi 4   = wceq 1540  {csn 4589  ⟨cop 4595   I cid 5532  ‘cfv 6511  0cc0 11068  ⟨“cs1 14560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-iota 6464  df-fv 6519  df-s1 14561

This theorem is referenced by:  s1eqd  14566  wrdl1exs1  14578  wrdl1s1  14579  ccats1pfxeqrex  14680  wrdind  14687  wrd2ind  14688  reuccatpfxs1lem  14711  reuccatpfxs1  14712  revs1  14730  vrmdval  18784  frgpup3lem  19707  vdegp1ci  29466  clwwlknonwwlknonb  30035  mrsubcv  35497  mrsubrn  35500  elmrsubrn  35507  mrsubvrs  35509  mvhval  35521