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Theorem s1rn 14494
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 14493 . . 3 (𝐴𝑉 → ⟨“𝐴”⟩ = {⟨0, 𝐴⟩})
21rneqd 5898 . 2 (𝐴𝑉 → ran ⟨“𝐴”⟩ = ran {⟨0, 𝐴⟩})
3 c0ex 11156 . . 3 0 ∈ V
43rnsnop 6181 . 2 ran {⟨0, 𝐴⟩} = {𝐴}
52, 4eqtrdi 2793 1 (𝐴𝑉 → ran ⟨“𝐴”⟩ = {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  {csn 4591  cop 4597  ran crn 5639  0cc0 11058  ⟨“cs1 14490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-mulcl 11120  ax-i2m1 11126
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6453  df-fun 6503  df-fv 6509  df-s1 14491
This theorem is referenced by:  cycpmco2f1  32015  cycpmco2rn  32016  mrsubvrs  34156
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