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Theorem op2nd 5697
Description: Extract the second member of an ordered pair. (Contributed by NM, 5-Oct-2004.)
Hypotheses
Ref Expression
op1st.1 A V
op1st.2 B V
Assertion
Ref Expression
op2nd (2nd ‘⟨A, B⟩) = B

Proof of Theorem op2nd
StepHypRef Expression
1 op1st.1 . . . 4 A V
2 op1st.2 . . . 4 B V
3 opexg 3938 . . . 4 ((A V B V) → ⟨A, B V)
41, 2, 3mp2an 404 . . 3 A, B V
5 2ndvalg 5693 . . 3 (⟨A, B V → (2nd ‘⟨A, B⟩) = ran {⟨A, B⟩})
64, 5ax-mp 7 . 2 (2nd ‘⟨A, B⟩) = ran {⟨A, B⟩}
71, 2op2nda 4732 . 2 ran {⟨A, B⟩} = B
86, 7eqtri 2042 1 (2nd ‘⟨A, B⟩) = B
Colors of variables: wff set class
Syntax hints:   = wceq 1228   wcel 1374  Vcvv 2535  {csn 3350  cop 3353   cuni 3554  ran crn 4273  cfv 4829  2nd c2nd 5689
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-un 4120
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-iota 4794  df-fun 4831  df-fv 4837  df-2nd 5691
This theorem is referenced by:  op2ndd  5699  op2ndg  5701  2ndval2  5706  fo2ndresm  5712  eloprabi  5745  fo2ndf  5771  f1o2ndf1  5772  genpelvu  6367  1pru  6406  ltexprlemelu  6436  recexprlemelu  6457
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