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Theorem op2nd 5716
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 3955 . . . 4 ((A V B V) → ⟨A, B V)
41, 2, 3mp2an 402 . . 3 A, B V
5 2ndvalg 5712 . . 3 (⟨A, B V → (2nd ‘⟨A, B⟩) = ran {⟨A, B⟩})
64, 5ax-mp 7 . 2 (2nd ‘⟨A, B⟩) = ran {⟨A, B⟩}
71, 2op2nda 4748 . 2 ran {⟨A, B⟩} = B
86, 7eqtri 2057 1 (2nd ‘⟨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  cfv 4845  2nd c2nd 5708
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-13 1401  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  ax-un 4136
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-sbc 2759  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-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fv 4853  df-2nd 5710
This theorem is referenced by:  op2ndd  5718  op2ndg  5720  2ndval2  5725  fo2ndresm  5731  eloprabi  5764  fo2ndf  5790  f1o2ndf1  5791  genpelvu  6495  1pru  6536  addnqpr1lemru  6538  addnqpr1lemil  6539  addnqpr1lemiu  6540  ltexprlemelu  6571  recexprlemelu  6593
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