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Theorem s1rn 14548
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 14547 . . 3 (𝐴𝑉 → ⟨“𝐴”⟩ = {⟨0, 𝐴⟩})
21rneqd 5937 . 2 (𝐴𝑉 → ran ⟨“𝐴”⟩ = ran {⟨0, 𝐴⟩})
3 c0ex 11207 . . 3 0 ∈ V
43rnsnop 6223 . 2 ran {⟨0, 𝐴⟩} = {𝐴}
52, 4eqtrdi 2788 1 (𝐴𝑉 → ran ⟨“𝐴”⟩ = {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  {csn 4628  cop 4634  ran crn 5677  0cc0 11109  ⟨“cs1 14544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-mulcl 11171  ax-i2m1 11177
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fv 6551  df-s1 14545
This theorem is referenced by:  cycpmco2f1  32278  cycpmco2rn  32279  mrsubvrs  34508
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