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Theorem op2nd 7125
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 7119 . 2 (2nd ‘⟨𝐴, 𝐵⟩) = ran {⟨𝐴, 𝐵⟩}
2 op1st.1 . . 3 𝐴 ∈ V
3 op1st.2 . . 3 𝐵 ∈ V
42, 3op2nda 5582 . 2 ran {⟨𝐴, 𝐵⟩} = 𝐵
51, 4eqtri 2648 1 (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wcel 1992  Vcvv 3191  {csn 4153  cop 4159   cuni 4407  ran crn 5080  cfv 5850  2nd c2nd 7115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-iota 5813  df-fun 5852  df-fv 5858  df-2nd 7117
This theorem is referenced by:  op2ndd  7127  op2ndg  7129  2ndval2  7134  fo2ndres  7141  eloprabi  7178  fo2ndf  7230  f1o2ndf1  7231  seqomlem1  7491  seqomlem2  7492  xpmapenlem  8072  fseqenlem2  8793  axdc4lem  9222  iunfo  9306  archnq  9747  om2uzrdg  12692  uzrdgsuci  12696  fsum2dlem  14424  fprod2dlem  14630  ruclem8  14886  ruclem11  14889  eucalglt  15217  idfu2nd  16453  idfucl  16457  cofu2nd  16461  cofucl  16464  xpccatid  16744  prf2nd  16761  curf2ndf  16803  yonedalem22  16834  gaid  17648  2ndcctbss  21163  upxp  21331  uptx  21333  txkgen  21360  cnheiborlem  22656  ovollb2lem  23158  ovolctb  23160  ovoliunlem2  23173  ovolshftlem1  23179  ovolscalem1  23183  ovolicc1  23186  wlkpwwlkf1ouspgr  26628  wlknwwlksnsur  26639  wlkwwlksur  26646  clwlksfoclwwlk  26823  ex-2nd  27150  cnnvs  27375  cnnvnm  27376  h2hsm  27672  h2hnm  27673  hhsssm  27955  hhssnm  27956  aciunf1lem  29295  eulerpartlemgvv  30211  eulerpartlemgh  30213  msubff1  31153  msubvrs  31157  br2ndeq  31367  poimirlem17  33044  heiborlem7  33234  heiborlem8  33235  dvhvaddass  35852  dvhlveclem  35863  diblss  35925  pellexlem5  36863  pellex  36865  dvnprodlem1  39454  hoicvr  40056  hoicvrrex  40064  ovn0lem  40073  ovnhoilem1  40109  ovnlecvr2  40118  ovolval5lem2  40161
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