Theorem op2nd 6315

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 4322	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ ∈ V)
4	1, 2, 3	mp2an 426	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
5		2ndvalg 6311	. . 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 5223	. 2 ⊢ ∪ ran {⟨𝐴, 𝐵⟩} = 𝐵
8	6, 7	eqtri 2251	1 ⊢ (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵

Syntax hints:   = wceq 1397   ∈ wcel 2201  Vcvv 2801  {csn 3670  ⟨cop 3673  ∪ cuni 3894  ran crn 4728  ‘cfv 5328  2^nd c2nd 6307

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 2203  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-un 4532

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-v 2803  df-sbc 3031  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-br 4090  df-opab 4152  df-mpt 4153  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-iota 5288  df-fun 5330  df-fv 5336  df-2nd 6309

This theorem is referenced by:  op2ndd  6317  op2ndg  6319  2ndval2  6324  fo2ndresm  6330  eloprabi  6366  fo2ndf  6397  f1o2ndf1  6398  xpmapenlem  7040  genpelvu  7738  nqprl  7776  1pru  7781  addnqprlemru  7783  addnqprlemfl  7784  addnqprlemfu  7785  mulnqprlemru  7799  mulnqprlemfl  7800  mulnqprlemfu  7801  ltnqpr  7818  ltnqpri  7819  ltexprlemelu  7824  recexprlemelu  7848  cauappcvgprlemm  7870  cauappcvgprlemopu  7873  cauappcvgprlemupu  7874  cauappcvgprlemdisj  7876  cauappcvgprlemloc  7877  cauappcvgprlemladdfu  7879  cauappcvgprlemladdru  7881  cauappcvgprlemladdrl  7882  cauappcvgprlem2  7885  caucvgprlemm  7893  caucvgprlemopu  7896  caucvgprlemupu  7897  caucvgprlemdisj  7899  caucvgprlemloc  7900  caucvgprlemladdfu  7902  caucvgprlem2  7905  caucvgprprlemelu  7911  caucvgprprlemmu  7920  caucvgprprlemexbt  7931  caucvgprprlem2  7935  suplocexprlemloc  7946  fsum2dlemstep  12018  fprod2dlemstep  12206  ctiunctlemfo  13083