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Theorem s1rn 13579
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 13578 . . 3 (𝐴𝑉 → ⟨“𝐴”⟩ = {⟨0, 𝐴⟩})
21rneqd 5491 . 2 (𝐴𝑉 → ran ⟨“𝐴”⟩ = ran {⟨0, 𝐴⟩})
3 c0ex 10236 . . 3 0 ∈ V
43rnsnop 5759 . 2 ran {⟨0, 𝐴⟩} = {𝐴}
52, 4syl6eq 2821 1 (𝐴𝑉 → ran ⟨“𝐴”⟩ = {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  {csn 4316  cop 4322  ran crn 5250  0cc0 10138  ⟨“cs1 13490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-mulcl 10200  ax-i2m1 10206
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fv 6039  df-s1 13498
This theorem is referenced by:  mrsubvrs  31757
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