Theorem s1val 14609

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 14607	. 2 ⊢ ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
2		fvi 6939	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ( I ‘𝐴) = 𝐴)
3	2	opeq2d 4837	. . 3 ⊢ (𝐴 ∈ 𝑉 → ⟨0, ( I ‘𝐴)⟩ = ⟨0, 𝐴⟩)
4	3	sneqd 4593	. 2 ⊢ (𝐴 ∈ 𝑉 → {⟨0, ( I ‘𝐴)⟩} = {⟨0, 𝐴⟩})
5	1, 4	eqtrid 2808	1 ⊢ (𝐴 ∈ 𝑉 → ⟨“𝐴”⟩ = {⟨0, 𝐴⟩})

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141  {csn 4581  ⟨cop 4587   I cid 5539  ‘cfv 6517  0cc0 11070  ⟨“cs1 14606

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-iota 6473  df-fun 6519  df-fv 6525  df-s1 14607

This theorem is referenced by:  s1rn  14610  s1cl  14613  s1dmALT  14620  s1fv  14621  s111  14626  repsw1  14793  s1co  14843  s2prop  14917  ofs1  14980  gsumws1  18855  uspgr1ewop  29395  usgr2v1e2w  29399  0wlkons1  30269  s1f1  33082  cshw1s2  33099  ofcs1  34802  signstf0  34826