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Theorem s1rn 14547
Description: The range of a singleton word. (Contributed by Mario Carneiro, 18-Jul-2016.)
Assertion
Ref Expression
s1rn (𝐴𝑉 → ran ⟨“𝐴”⟩ = {𝐴})

Proof of Theorem s1rn
StepHypRef Expression
1 s1val 14546 . . 3 (𝐴𝑉 → ⟨“𝐴”⟩ = {⟨0, 𝐴⟩})
21rneqd 5928 . 2 (𝐴𝑉 → ran ⟨“𝐴”⟩ = ran {⟨0, 𝐴⟩})
3 c0ex 11206 . . 3 0 ∈ V
43rnsnop 6214 . 2 ran {⟨0, 𝐴⟩} = {𝐴}
52, 4eqtrdi 2780 1 (𝐴𝑉 → ran ⟨“𝐴”⟩ = {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  {csn 4621  cop 4627  ran crn 5668  0cc0 11107  ⟨“cs1 14543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-mulcl 11169  ax-i2m1 11175
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-iota 6486  df-fun 6536  df-fv 6542  df-s1 14544
This theorem is referenced by:  cycpmco2f1  32754  cycpmco2rn  32755  mrsubvrs  35004
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