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Theorem rnsnop 6078
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 6075 . 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 1530  wcel 2107  Vcvv 3499  {csn 4563  cop 4569  ran crn 5554
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-br 5063  df-opab 5125  df-xp 5559  df-rel 5560  df-cnv 5561  df-dm 5563  df-rn 5564
This theorem is referenced by:  op2nda  6082  fpr  6911  en1  8568  fodomfi  8789  dcomex  9861  s1rn  13946  axlowdimlem13  26657  ex-rn  28136  ex-ima  28138  ptrest  34761  poimirlem3  34765  gidsn  35101  zrdivrng  35102
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