Theorem op2nd 6305

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 4318	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ ∈ V)
4	1, 2, 3	mp2an 426	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
5		2ndvalg 6301	. . 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 5219	. 2 ⊢ ∪ ran {⟨𝐴, 𝐵⟩} = 𝐵
8	6, 7	eqtri 2250	1 ⊢ (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵

Syntax hints:   = wceq 1395   ∈ wcel 2200  Vcvv 2800  {csn 3667  ⟨cop 3670  ∪ cuni 3891  ran crn 4724  ‘cfv 5324  2^nd c2nd 6297

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-sbc 3030  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-iota 5284  df-fun 5326  df-fv 5332  df-2nd 6299

This theorem is referenced by:  op2ndd  6307  op2ndg  6309  2ndval2  6314  fo2ndresm  6320  eloprabi  6356  fo2ndf  6387  f1o2ndf1  6388  xpmapenlem  7030  genpelvu  7726  nqprl  7764  1pru  7769  addnqprlemru  7771  addnqprlemfl  7772  addnqprlemfu  7773  mulnqprlemru  7787  mulnqprlemfl  7788  mulnqprlemfu  7789  ltnqpr  7806  ltnqpri  7807  ltexprlemelu  7812  recexprlemelu  7836  cauappcvgprlemm  7858  cauappcvgprlemopu  7861  cauappcvgprlemupu  7862  cauappcvgprlemdisj  7864  cauappcvgprlemloc  7865  cauappcvgprlemladdfu  7867  cauappcvgprlemladdru  7869  cauappcvgprlemladdrl  7870  cauappcvgprlem2  7873  caucvgprlemm  7881  caucvgprlemopu  7884  caucvgprlemupu  7885  caucvgprlemdisj  7887  caucvgprlemloc  7888  caucvgprlemladdfu  7890  caucvgprlem2  7893  caucvgprprlemelu  7899  caucvgprprlemmu  7908  caucvgprprlemexbt  7919  caucvgprprlem2  7923  suplocexprlemloc  7934  fsum2dlemstep  11988  fprod2dlemstep  12176  ctiunctlemfo  13053