Theorem s1eq 11147

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 5626	. . . 4 ⊢ (𝐴 = 𝐵 → ( I ‘𝐴) = ( I ‘𝐵))
2	1	opeq2d 3863	. . 3 ⊢ (𝐴 = 𝐵 → ⟨0, ( I ‘𝐴)⟩ = ⟨0, ( I ‘𝐵)⟩)
3	2	sneqd 3679	. 2 ⊢ (𝐴 = 𝐵 → {⟨0, ( I ‘𝐴)⟩} = {⟨0, ( I ‘𝐵)⟩})
4		df-s1 11144	. 2 ⊢ ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
5		df-s1 11144	. 2 ⊢ ⟨“𝐵”⟩ = {⟨0, ( I ‘𝐵)⟩}
6	3, 4, 5	3eqtr4g 2287	1 ⊢ (𝐴 = 𝐵 → ⟨“𝐴”⟩ = ⟨“𝐵”⟩)

Syntax hints:   → wi 4   = wceq 1395  {csn 3666  ⟨cop 3669   I cid 4378  ‘cfv 5317  0cc0 7995  ⟨“cs1 11143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-iota 5277  df-fv 5325  df-s1 11144

This theorem is referenced by:  s1eqd  11148  wrdl1exs1  11157  wrdl1s1  11158  ccats1pfxeqrex  11242  wrdind  11249  wrd2ind  11250  reuccatpfxs1lem  11273  reuccatpfxs1  11274