Theorem s1val 14570

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 14568	. 2 ⊢ ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
2		fvi 6940	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ( I ‘𝐴) = 𝐴)
3	2	opeq2d 4847	. . 3 ⊢ (𝐴 ∈ 𝑉 → ⟨0, ( I ‘𝐴)⟩ = ⟨0, 𝐴⟩)
4	3	sneqd 4604	. 2 ⊢ (𝐴 ∈ 𝑉 → {⟨0, ( I ‘𝐴)⟩} = {⟨0, 𝐴⟩})
5	1, 4	eqtrid 2777	1 ⊢ (𝐴 ∈ 𝑉 → ⟨“𝐴”⟩ = {⟨0, 𝐴⟩})

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  {csn 4592  ⟨cop 4598   I cid 5535  ‘cfv 6514  0cc0 11075  ⟨“cs1 14567

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

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-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-iota 6467  df-fun 6516  df-fv 6522  df-s1 14568

This theorem is referenced by:  s1rn  14571  s1cl  14574  s1dmALT  14581  s1fv  14582  s111  14587  repsw1  14755  s1co  14806  s2prop  14880  ofs1  14943  gsumws1  18772  uspgr1ewop  29182  usgr2v1e2w  29186  0wlkons1  30057  s1f1  32871  cshw1s2  32889  ofcs1  34542  signstf0  34566