Theorem s1eq 14377

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 6811	. . . 4 ⊢ (𝐴 = 𝐵 → ( I ‘𝐴) = ( I ‘𝐵))
2	1	opeq2d 4822	. . 3 ⊢ (𝐴 = 𝐵 → ⟨0, ( I ‘𝐴)⟩ = ⟨0, ( I ‘𝐵)⟩)
3	2	sneqd 4583	. 2 ⊢ (𝐴 = 𝐵 → {⟨0, ( I ‘𝐴)⟩} = {⟨0, ( I ‘𝐵)⟩})
4		df-s1 14373	. 2 ⊢ ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
5		df-s1 14373	. 2 ⊢ ⟨“𝐵”⟩ = {⟨0, ( I ‘𝐵)⟩}
6	3, 4, 5	3eqtr4g 2802	1 ⊢ (𝐴 = 𝐵 → ⟨“𝐴”⟩ = ⟨“𝐵”⟩)

Syntax hints:   → wi 4   = wceq 1540  {csn 4571  ⟨cop 4577   I cid 5506  ‘cfv 6465  0cc0 10944  ⟨“cs1 14372

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2708

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2715  df-cleq 2729  df-clel 2815  df-rab 3405  df-v 3443  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4268  df-if 4472  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4851  df-br 5088  df-iota 6417  df-fv 6473  df-s1 14373

This theorem is referenced by:  s1eqd  14378  wrdl1exs1  14390  wrdl1s1  14391  ccats1pfxeqrex  14500  wrdind  14507  wrd2ind  14508  reuccatpfxs1lem  14531  reuccatpfxs1  14532  revs1  14550  vrmdval  18565  frgpup3lem  19451  vdegp1ci  28014  clwwlknonwwlknonb  28579  mrsubcv  33577  mrsubrn  33580  elmrsubrn  33587  mrsubvrs  33589  mvhval  33601