MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  op2nd Structured version   Visualization version   GIF version

Theorem op2nd 7342
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 7336 . 2 (2nd ‘⟨𝐴, 𝐵⟩) = ran {⟨𝐴, 𝐵⟩}
2 op1st.1 . . 3 𝐴 ∈ V
3 op1st.2 . . 3 𝐵 ∈ V
42, 3op2nda 5781 . 2 ran {⟨𝐴, 𝐵⟩} = 𝐵
51, 4eqtri 2782 1 (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1632  wcel 2139  Vcvv 3340  {csn 4321  cop 4327   cuni 4588  ran crn 5267  cfv 6049  2nd c2nd 7332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fv 6057  df-2nd 7334
This theorem is referenced by:  op2ndd  7344  op2ndg  7346  2ndval2  7351  fo2ndres  7360  eloprabi  7400  fo2ndf  7452  f1o2ndf1  7453  seqomlem1  7714  seqomlem2  7715  xpmapenlem  8292  fseqenlem2  9038  axdc4lem  9469  iunfo  9553  archnq  9994  om2uzrdg  12949  uzrdgsuci  12953  fsum2dlem  14700  fprod2dlem  14909  ruclem8  15165  ruclem11  15168  eucalglt  15500  idfu2nd  16738  idfucl  16742  cofu2nd  16746  cofucl  16749  xpccatid  17029  prf2nd  17046  curf2ndf  17088  yonedalem22  17119  gaid  17932  2ndcctbss  21460  upxp  21628  uptx  21630  txkgen  21657  cnheiborlem  22954  ovollb2lem  23456  ovolctb  23458  ovoliunlem2  23471  ovolshftlem1  23477  ovolscalem1  23481  ovolicc1  23484  wlkpwwlkf1ouspgr  26988  wlknwwlksnsur  26999  wlkwwlksur  27006  clwlkclwwlkfo  27132  clwlksfoclwwlkOLD  27217  ex-2nd  27613  cnnvs  27844  cnnvnm  27845  h2hsm  28141  h2hnm  28142  hhsssm  28424  hhssnm  28425  aciunf1lem  29771  eulerpartlemgvv  30747  eulerpartlemgh  30749  msubff1  31760  msubvrs  31764  poimirlem17  33739  heiborlem7  33929  heiborlem8  33930  dvhvaddass  36888  dvhlveclem  36899  diblss  36961  pellexlem5  37899  pellex  37901  dvnprodlem1  40664  hoicvr  41268  hoicvrrex  41276  ovn0lem  41285  ovnhoilem1  41321  ovnlecvr2  41330  ovolval5lem2  41373
  Copyright terms: Public domain W3C validator