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Theorem op2nd 7680
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
StepHypRef Expression
1 2ndval 7674 . 2 (2nd ‘⟨𝐴, 𝐵⟩) = ran {⟨𝐴, 𝐵⟩}
2 op1st.1 . . 3 𝐴 ∈ V
3 op1st.2 . . 3 𝐵 ∈ V
42, 3op2nda 6052 . 2 ran {⟨𝐴, 𝐵⟩} = 𝐵
51, 4eqtri 2821 1 (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  wcel 2111  Vcvv 3441  {csn 4525  cop 4531   cuni 4800  ran crn 5520  cfv 6324  2nd c2nd 7670
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-iota 6283  df-fun 6326  df-fv 6332  df-2nd 7672
This theorem is referenced by:  op2ndd  7682  op2ndg  7684  2ndval2  7689  fo2ndres  7698  opreuopreu  7716  eloprabi  7743  fo2ndf  7800  f1o2ndf1  7801  seqomlem1  8069  seqomlem2  8070  xpmapenlem  8668  fseqenlem2  9436  axdc4lem  9866  iunfo  9950  archnq  10391  om2uzrdg  13319  uzrdgsuci  13323  fsum2dlem  15117  fprod2dlem  15326  ruclem8  15582  ruclem11  15585  eucalglt  15919  idfu2nd  17139  idfucl  17143  cofu2nd  17147  cofucl  17150  xpccatid  17430  prf2nd  17447  curf2ndf  17489  yonedalem22  17520  gaid  18421  2ndcctbss  22060  upxp  22228  uptx  22230  txkgen  22257  cnheiborlem  23559  ovollb2lem  24092  ovolctb  24094  ovoliunlem2  24107  ovolshftlem1  24113  ovolscalem1  24117  ovolicc1  24120  addsqnreup  26027  2sqreuop  26046  2sqreuopnn  26047  2sqreuoplt  26048  2sqreuopltb  26049  2sqreuopnnlt  26050  2sqreuopnnltb  26051  wlkswwlksf1o  27665  clwlkclwwlkfo  27794  ex-2nd  28230  cnnvs  28463  cnnvnm  28464  h2hsm  28758  h2hnm  28759  hhsssm  29041  hhssnm  29042  2ndimaxp  30409  2ndresdju  30411  aciunf1lem  30425  gsumpart  30740  eulerpartlemgvv  31744  eulerpartlemgh  31746  satfv0fvfmla0  32773  sategoelfvb  32779  prv1n  32791  msubff1  32916  msubvrs  32920  poimirlem17  35074  heiborlem7  35255  heiborlem8  35256  dvhvaddass  38393  dvhlveclem  38404  diblss  38466  pellexlem5  39772  pellex  39774  dvnprodlem1  42586  hoicvr  43185  hoicvrrex  43193  ovn0lem  43202  ovnhoilem1  43238  ovnlecvr2  43247  ovolval5lem2  43290
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