Theorem op2nd 6340

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 4343	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ ∈ V)
4	1, 2, 3	mp2an 426	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
5		2ndvalg 6336	. . 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 5246	. 2 ⊢ ∪ ran {⟨𝐴, 𝐵⟩} = 𝐵
8	6, 7	eqtri 2253	1 ⊢ (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵

Syntax hints:   = wceq 1398   ∈ wcel 2203  Vcvv 2812  {csn 3688  ⟨cop 3691  ∪ cuni 3913  ran crn 4749  ‘cfv 5351  2^nd c2nd 6332

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-sbc 3042  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-iota 5311  df-fun 5353  df-fv 5359  df-2nd 6334

This theorem is referenced by:  op2ndd  6342  op2ndg  6344  2ndval2  6349  fo2ndresm  6355  eloprabi  6391  fo2ndf  6422  f1o2ndf1  6423  xpmapenlem  7101  genpelvu  7824  nqprl  7862  1pru  7867  addnqprlemru  7869  addnqprlemfl  7870  addnqprlemfu  7871  mulnqprlemru  7885  mulnqprlemfl  7886  mulnqprlemfu  7887  ltnqpr  7904  ltnqpri  7905  ltexprlemelu  7910  recexprlemelu  7934  cauappcvgprlemm  7956  cauappcvgprlemopu  7959  cauappcvgprlemupu  7960  cauappcvgprlemdisj  7962  cauappcvgprlemloc  7963  cauappcvgprlemladdfu  7965  cauappcvgprlemladdru  7967  cauappcvgprlemladdrl  7968  cauappcvgprlem2  7971  caucvgprlemm  7979  caucvgprlemopu  7982  caucvgprlemupu  7983  caucvgprlemdisj  7985  caucvgprlemloc  7986  caucvgprlemladdfu  7988  caucvgprlem2  7991  caucvgprprlemelu  7997  caucvgprprlemmu  8006  caucvgprprlemexbt  8017  caucvgprprlem2  8021  suplocexprlemloc  8032  fsum2dlemstep  12113  fprod2dlemstep  12301  ctiunctlemfo  13179