Theorem op2nd 6375

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

Step	Hyp	Ref	Expression
1		op1st.1	. . . 4 ⊢ 𝐴 ∈ V
2		op1st.2	. . . 4 ⊢ 𝐵 ∈ V
3		opexg 4366	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ ∈ V)
4	1, 2, 3	mp2an 430	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
5		2ndvalg 6371	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ V → (2nd ‘⟨𝐴, 𝐵⟩) = ∪ ran {⟨𝐴, 𝐵⟩})
6	4, 5	ax-mp 5	. 2 ⊢ (2nd ‘⟨𝐴, 𝐵⟩) = ∪ ran {⟨𝐴, 𝐵⟩}
7	1, 2	op2nda 5270	. 2 ⊢ ∪ ran {⟨𝐴, 𝐵⟩} = 𝐵
8	6, 7	eqtri 2259	1 ⊢ (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵

Syntax hints:   = wceq 1402   ∈ wcel 2209  Vcvv 2821  {csn 3708  ⟨cop 3711  ∪ cuni 3933  ran crn 4773  ‘cfv 5375  2^nd c2nd 6367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-iota 5335  df-fun 5377  df-fv 5383  df-2nd 6369

This theorem is referenced by:  op2ndd  6377  op2ndg  6379  2ndval2  6384  fo2ndresm  6390  eloprabi  6426  fo2ndf  6457  f1o2ndf1  6458  xpmapenlem  7143  genpelvu  7874  nqprl  7912  1pru  7917  addnqprlemru  7919  addnqprlemfl  7920  addnqprlemfu  7921  mulnqprlemru  7935  mulnqprlemfl  7936  mulnqprlemfu  7937  ltnqpr  7954  ltnqpri  7955  ltexprlemelu  7960  recexprlemelu  7984  cauappcvgprlemm  8006  cauappcvgprlemopu  8009  cauappcvgprlemupu  8010  cauappcvgprlemdisj  8012  cauappcvgprlemloc  8013  cauappcvgprlemladdfu  8015  cauappcvgprlemladdru  8017  cauappcvgprlemladdrl  8018  cauappcvgprlem2  8021  caucvgprlemm  8029  caucvgprlemopu  8032  caucvgprlemupu  8033  caucvgprlemdisj  8035  caucvgprlemloc  8036  caucvgprlemladdfu  8038  caucvgprlem2  8041  caucvgprprlemelu  8047  caucvgprprlemmu  8056  caucvgprprlemexbt  8067  caucvgprprlem2  8071  suplocexprlemloc  8082  fsum2dlemstep  12184  fprod2dlemstep  12372  ctiunctlemfo  13313