Theorem s1val 14508

Description: Value of a singleton word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)

Assertion

Ref	Expression
s1val	⊢ (𝐴 ∈ 𝑉 → ⟨“𝐴”⟩ = {⟨0, 𝐴⟩})

Proof of Theorem s1val

Step	Hyp	Ref	Expression
1		df-s1 14506	. 2 ⊢ ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
2		fvi 6904	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ( I ‘𝐴) = 𝐴)
3	2	opeq2d 4831	. . 3 ⊢ (𝐴 ∈ 𝑉 → ⟨0, ( I ‘𝐴)⟩ = ⟨0, 𝐴⟩)
4	3	sneqd 4587	. 2 ⊢ (𝐴 ∈ 𝑉 → {⟨0, ( I ‘𝐴)⟩} = {⟨0, 𝐴⟩})
5	1, 4	eqtrid 2780	1 ⊢ (𝐴 ∈ 𝑉 → ⟨“𝐴”⟩ = {⟨0, 𝐴⟩})

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  {csn 4575  ⟨cop 4581   I cid 5513  ‘cfv 6486  0cc0 11013  ⟨“cs1 14505

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-iota 6442  df-fun 6488  df-fv 6494  df-s1 14506

This theorem is referenced by:  s1rn  14509  s1cl  14512  s1dmALT  14519  s1fv  14520  s111  14525  repsw1  14692  s1co  14742  s2prop  14816  ofs1  14879  gsumws1  18748  uspgr1ewop  29228  usgr2v1e2w  29232  0wlkons1  30103  s1f1  32931  cshw1s2  32948  ofcs1  34578  signstf0  34602