Theorem s1eq 14536

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 6842	. . . 4 ⊢ (𝐴 = 𝐵 → ( I ‘𝐴) = ( I ‘𝐵))
2	1	opeq2d 4838	. . 3 ⊢ (𝐴 = 𝐵 → ⟨0, ( I ‘𝐴)⟩ = ⟨0, ( I ‘𝐵)⟩)
3	2	sneqd 4594	. 2 ⊢ (𝐴 = 𝐵 → {⟨0, ( I ‘𝐴)⟩} = {⟨0, ( I ‘𝐵)⟩})
4		df-s1 14532	. 2 ⊢ ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
5		df-s1 14532	. 2 ⊢ ⟨“𝐵”⟩ = {⟨0, ( I ‘𝐵)⟩}
6	3, 4, 5	3eqtr4g 2797	1 ⊢ (𝐴 = 𝐵 → ⟨“𝐴”⟩ = ⟨“𝐵”⟩)

Syntax hints:   → wi 4   = wceq 1542  {csn 4582  ⟨cop 4588   I cid 5526  ‘cfv 6500  0cc0 11038  ⟨“cs1 14531

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6456  df-fv 6508  df-s1 14532

This theorem is referenced by:  s1eqd  14537  wrdl1exs1  14549  wrdl1s1  14550  ccats1pfxeqrex  14650  wrdind  14657  wrd2ind  14658  reuccatpfxs1lem  14681  reuccatpfxs1  14682  revs1  14700  chninf  18570  vrmdval  18794  frgpup3lem  19718  vdegp1ci  29624  clwwlknonwwlknonb  30193  mrsubcv  35723  mrsubrn  35726  elmrsubrn  35733  mrsubvrs  35735  mvhval  35747