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Theorem rnsnop 5776
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 5773 . 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 1632  wcel 2139  Vcvv 3340  {csn 4321  cop 4327  ran crn 5267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-dm 5276  df-rn 5277
This theorem is referenced by:  op2nda  5781  fpr  6585  en1  8190  fodomfi  8406  dcomex  9481  s1rn  13589  axlowdimlem13  26054  ex-rn  27629  ex-ima  27631  ptrest  33739  poimirlem3  33743  gidsn  34082  zrdivrng  34083
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