Theorem s1eq 14616

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 6875	. . . 4 ⊢ (𝐴 = 𝐵 → ( I ‘𝐴) = ( I ‘𝐵))
2	1	opeq2d 4856	. . 3 ⊢ (𝐴 = 𝐵 → ⟨0, ( I ‘𝐴)⟩ = ⟨0, ( I ‘𝐵)⟩)
3	2	sneqd 4613	. 2 ⊢ (𝐴 = 𝐵 → {⟨0, ( I ‘𝐴)⟩} = {⟨0, ( I ‘𝐵)⟩})
4		df-s1 14612	. 2 ⊢ ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
5		df-s1 14612	. 2 ⊢ ⟨“𝐵”⟩ = {⟨0, ( I ‘𝐵)⟩}
6	3, 4, 5	3eqtr4g 2795	1 ⊢ (𝐴 = 𝐵 → ⟨“𝐴”⟩ = ⟨“𝐵”⟩)

Syntax hints:   → wi 4   = wceq 1540  {csn 4601  ⟨cop 4607   I cid 5547  ‘cfv 6530  0cc0 11127  ⟨“cs1 14611

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 2007  ax-8 2110  ax-9 2118  ax-ext 2707

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 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-iota 6483  df-fv 6538  df-s1 14612

This theorem is referenced by:  s1eqd  14617  wrdl1exs1  14629  wrdl1s1  14630  ccats1pfxeqrex  14731  wrdind  14738  wrd2ind  14739  reuccatpfxs1lem  14762  reuccatpfxs1  14763  revs1  14781  vrmdval  18833  frgpup3lem  19756  vdegp1ci  29464  clwwlknonwwlknonb  30033  mrsubcv  35478  mrsubrn  35481  elmrsubrn  35488  mrsubvrs  35490  mvhval  35502