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Theorem rnsnop 6214
Description: The range of a singleton of an ordered pair is the singleton of the second member. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
Hypothesis
Ref Expression
cnvsn.1 𝐴 ∈ V
Assertion
Ref Expression
rnsnop ran {⟨𝐴, 𝐵⟩} = {𝐵}

Proof of Theorem rnsnop
StepHypRef Expression
1 cnvsn.1 . 2 𝐴 ∈ V
2 rnsnopg 6211 . 2 (𝐴 ∈ V → ran {⟨𝐴, 𝐵⟩} = {𝐵})
31, 2ax-mp 5 1 ran {⟨𝐴, 𝐵⟩} = {𝐵}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wcel 2098  Vcvv 3466  {csn 4621  cop 4627  ran crn 5668
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-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
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-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  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-br 5140  df-opab 5202  df-xp 5673  df-rel 5674  df-cnv 5675  df-dm 5677  df-rn 5678
This theorem is referenced by:  op2nda  6218  fpr  7145  en1  9018  en1OLD  9019  fodomfi  9322  dcomex  10439  s1rn  14547  axlowdimlem13  28684  ex-rn  30165  ex-ima  30167  ptrest  36981  poimirlem3  36985  gidsn  37314  zrdivrng  37315
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