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Theorem op2nda 6082
Description: Extract the second member of an ordered pair. (See op1sta 6079 to extract the first member, op2ndb 6081 for an alternate version, and op2nd 7692 for the preferred version.) (Contributed by NM, 17-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Hypotheses
Ref Expression
cnvsn.1 𝐴 ∈ V
cnvsn.2 𝐵 ∈ V
Assertion
Ref Expression
op2nda ran {⟨𝐴, 𝐵⟩} = 𝐵

Proof of Theorem op2nda
StepHypRef Expression
1 cnvsn.1 . . . 4 𝐴 ∈ V
21rnsnop 6078 . . 3 ran {⟨𝐴, 𝐵⟩} = {𝐵}
32unieqi 4845 . 2 ran {⟨𝐴, 𝐵⟩} = {𝐵}
4 cnvsn.2 . . 3 𝐵 ∈ V
54unisn 4852 . 2 {𝐵} = 𝐵
63, 5eqtri 2848 1 ran {⟨𝐴, 𝐵⟩} = 𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1530  wcel 2107  Vcvv 3499  {csn 4563  cop 4569   cuni 4836  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-rex 3148  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-uni 4837  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:  elxp4  7618  elxp5  7619  op2nd  7692  fo2nd  7704  f2ndres  7708  ixpsnf1o  8494  xpassen  8603  xpdom2  8604
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