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Theorem op2nda 5782
Description: Extract the second member of an ordered pair. (See op1sta 5778 to extract the first member, op2ndb 5781 for an alternate version, and op2nd 7344 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 5777 . . 3 ran {⟨𝐴, 𝐵⟩} = {𝐵}
32unieqi 4598 . 2 ran {⟨𝐴, 𝐵⟩} = {𝐵}
4 cnvsn.2 . . 3 𝐵 ∈ V
54unisn 4604 . 2 {𝐵} = 𝐵
63, 5eqtri 2783 1 ran {⟨𝐴, 𝐵⟩} = 𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1632  wcel 2140  Vcvv 3341  {csn 4322  cop 4328   cuni 4589  ran crn 5268
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 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056
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 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-rex 3057  df-rab 3060  df-v 3343  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-xp 5273  df-rel 5274  df-cnv 5275  df-dm 5277  df-rn 5278
This theorem is referenced by:  elxp4  7277  elxp5  7278  op2nd  7344  fo2nd  7356  f2ndres  7360  ixpsnf1o  8117  xpassen  8222  xpdom2  8223
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