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Theorem op2nd 5774
 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 3964 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ ∈ V)
41, 2, 3mp2an 402 . . 3 𝐴, 𝐵⟩ ∈ V
5 2ndvalg 5770 . . 3 (⟨𝐴, 𝐵⟩ ∈ V → (2nd ‘⟨𝐴, 𝐵⟩) = ran {⟨𝐴, 𝐵⟩})
64, 5ax-mp 7 . 2 (2nd ‘⟨𝐴, 𝐵⟩) = ran {⟨𝐴, 𝐵⟩}
71, 2op2nda 4805 . 2 ran {⟨𝐴, 𝐵⟩} = 𝐵
86, 7eqtri 2060 1 (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵
 Colors of variables: wff set class Syntax hints:   = wceq 1243   ∈ wcel 1393  Vcvv 2557  {csn 3375  ⟨cop 3378  ∪ cuni 3580  ran crn 4346  ‘cfv 4902  2nd c2nd 5766 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fv 4910  df-2nd 5768 This theorem is referenced by:  op2ndd  5776  op2ndg  5778  2ndval2  5783  fo2ndresm  5789  eloprabi  5822  fo2ndf  5848  f1o2ndf1  5849  genpelvu  6611  nqprl  6649  1pru  6654  addnqprlemru  6656  addnqprlemfl  6657  addnqprlemfu  6658  mulnqprlemru  6672  mulnqprlemfl  6673  mulnqprlemfu  6674  ltnqpr  6691  ltnqpri  6692  ltexprlemelu  6697  recexprlemelu  6721  cauappcvgprlemm  6743  cauappcvgprlemopu  6746  cauappcvgprlemupu  6747  cauappcvgprlemdisj  6749  cauappcvgprlemloc  6750  cauappcvgprlemladdfu  6752  cauappcvgprlemladdru  6754  cauappcvgprlemladdrl  6755  cauappcvgprlem2  6758  caucvgprlemm  6766  caucvgprlemopu  6769  caucvgprlemupu  6770  caucvgprlemdisj  6772  caucvgprlemloc  6773  caucvgprlemladdfu  6775  caucvgprlem2  6778  caucvgprprlemelu  6784  caucvgprprlemmu  6793  caucvgprprlemexbt  6804  caucvgprprlem2  6808
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