Theorem op2nd 6343

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 4346	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ ∈ V)
4	1, 2, 3	mp2an 426	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
5		2ndvalg 6339	. . 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 5249	. 2 ⊢ ∪ ran {⟨𝐴, 𝐵⟩} = 𝐵
8	6, 7	eqtri 2255	1 ⊢ (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵

Syntax hints:   = wceq 1398   ∈ wcel 2205  Vcvv 2815  {csn 3691  ⟨cop 3694  ∪ cuni 3916  ran crn 4752  ‘cfv 5354  2^nd c2nd 6335

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3045  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-iota 5314  df-fun 5356  df-fv 5362  df-2nd 6337

This theorem is referenced by:  op2ndd  6345  op2ndg  6347  2ndval2  6352  fo2ndresm  6358  eloprabi  6394  fo2ndf  6425  f1o2ndf1  6426  xpmapenlem  7104  genpelvu  7833  nqprl  7871  1pru  7876  addnqprlemru  7878  addnqprlemfl  7879  addnqprlemfu  7880  mulnqprlemru  7894  mulnqprlemfl  7895  mulnqprlemfu  7896  ltnqpr  7913  ltnqpri  7914  ltexprlemelu  7919  recexprlemelu  7943  cauappcvgprlemm  7965  cauappcvgprlemopu  7968  cauappcvgprlemupu  7969  cauappcvgprlemdisj  7971  cauappcvgprlemloc  7972  cauappcvgprlemladdfu  7974  cauappcvgprlemladdru  7976  cauappcvgprlemladdrl  7977  cauappcvgprlem2  7980  caucvgprlemm  7988  caucvgprlemopu  7991  caucvgprlemupu  7992  caucvgprlemdisj  7994  caucvgprlemloc  7995  caucvgprlemladdfu  7997  caucvgprlem2  8000  caucvgprprlemelu  8006  caucvgprprlemmu  8015  caucvgprprlemexbt  8026  caucvgprprlem2  8030  suplocexprlemloc  8041  fsum2dlemstep  12128  fprod2dlemstep  12316  ctiunctlemfo  13211