Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  op2nda Structured version   GIF version

Theorem op2nda 4748
 Description: Extract the second member of an ordered pair. (See op1sta 4745 to extract the first member and op2ndb 4747 for an alternate 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 4744 . . 3 ran {⟨A, B⟩} = {B}
32unieqi 3581 . 2 ran {⟨A, B⟩} = {B}
4 cnvsn.2 . . 3 B V
54unisn 3587 . 2 {B} = B
63, 5eqtri 2057 1 ran {⟨A, B⟩} = B
 Colors of variables: wff set class Syntax hints:   = wceq 1242   ∈ wcel 1390  Vcvv 2551  {csn 3367  ⟨cop 3370  ∪ cuni 3571  ran crn 4289 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299 This theorem is referenced by:  elxp4  4751  elxp5  4752  op2nd  5716  fo2nd  5727  f2ndres  5729  xpassen  6240  xpdom2  6241
 Copyright terms: Public domain W3C validator