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Theorem s1rn 14576
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 14575 . . 3 (𝐴𝑉 → ⟨“𝐴”⟩ = {⟨0, 𝐴⟩})
21rneqd 5935 . 2 (𝐴𝑉 → ran ⟨“𝐴”⟩ = ran {⟨0, 𝐴⟩})
3 c0ex 11233 . . 3 0 ∈ V
43rnsnop 6223 . 2 ran {⟨0, 𝐴⟩} = {𝐴}
52, 4eqtrdi 2784 1 (𝐴𝑉 → ran ⟨“𝐴”⟩ = {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  {csn 4625  cop 4631  ran crn 5674  0cc0 11133  ⟨“cs1 14572
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2167  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pr 5424  ax-1cn 11191  ax-icn 11192  ax-addcl 11193  ax-mulcl 11195  ax-i2m1 11201
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5144  df-opab 5206  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-iota 6495  df-fun 6545  df-fv 6551  df-s1 14573
This theorem is referenced by:  cycpmco2f1  32840  cycpmco2rn  32841  mrsubvrs  35127
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