Theorem op2nd 6310

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 4320	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ ∈ V)
4	1, 2, 3	mp2an 426	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
5		2ndvalg 6306	. . 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 5221	. 2 ⊢ ∪ ran {⟨𝐴, 𝐵⟩} = 𝐵
8	6, 7	eqtri 2252	1 ⊢ (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵

Syntax hints:   = wceq 1397   ∈ wcel 2202  Vcvv 2802  {csn 3669  ⟨cop 3672  ∪ cuni 3893  ran crn 4726  ‘cfv 5326  2^nd c2nd 6302

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fv 5334  df-2nd 6304

This theorem is referenced by:  op2ndd  6312  op2ndg  6314  2ndval2  6319  fo2ndresm  6325  eloprabi  6361  fo2ndf  6392  f1o2ndf1  6393  xpmapenlem  7035  genpelvu  7733  nqprl  7771  1pru  7776  addnqprlemru  7778  addnqprlemfl  7779  addnqprlemfu  7780  mulnqprlemru  7794  mulnqprlemfl  7795  mulnqprlemfu  7796  ltnqpr  7813  ltnqpri  7814  ltexprlemelu  7819  recexprlemelu  7843  cauappcvgprlemm  7865  cauappcvgprlemopu  7868  cauappcvgprlemupu  7869  cauappcvgprlemdisj  7871  cauappcvgprlemloc  7872  cauappcvgprlemladdfu  7874  cauappcvgprlemladdru  7876  cauappcvgprlemladdrl  7877  cauappcvgprlem2  7880  caucvgprlemm  7888  caucvgprlemopu  7891  caucvgprlemupu  7892  caucvgprlemdisj  7894  caucvgprlemloc  7895  caucvgprlemladdfu  7897  caucvgprlem2  7900  caucvgprprlemelu  7906  caucvgprprlemmu  7915  caucvgprprlemexbt  7926  caucvgprprlem2  7930  suplocexprlemloc  7941  fsum2dlemstep  11997  fprod2dlemstep  12185  ctiunctlemfo  13062