Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  op2nda Structured version   Visualization version   GIF version

Theorem op2nda 5584
 Description: Extract the second member of an ordered pair. (See op1sta 5581 to extract the first member, op2ndb 5583 for an alternate version, and op2nd 7129 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 5580 . . 3 ran {⟨𝐴, 𝐵⟩} = {𝐵}
32unieqi 4416 . 2 ran {⟨𝐴, 𝐵⟩} = {𝐵}
4 cnvsn.2 . . 3 𝐵 ∈ V
54unisn 4422 . 2 {𝐵} = 𝐵
63, 5eqtri 2643 1 ran {⟨𝐴, 𝐵⟩} = 𝐵
 Colors of variables: wff setvar class Syntax hints:   = wceq 1480   ∈ wcel 1987  Vcvv 3189  {csn 4153  ⟨cop 4159  ∪ cuni 4407  ran crn 5080 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-xp 5085  df-rel 5086  df-cnv 5087  df-dm 5089  df-rn 5090 This theorem is referenced by:  elxp4  7064  elxp5  7065  op2nd  7129  fo2nd  7141  f2ndres  7143  ixpsnf1o  7900  xpassen  8006  xpdom2  8007
 Copyright terms: Public domain W3C validator