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Theorem op1st 7686
Description: Extract the first 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
op1st (1st ‘⟨𝐴, 𝐵⟩) = 𝐴

Proof of Theorem op1st
StepHypRef Expression
1 1stval 7680 . 2 (1st ‘⟨𝐴, 𝐵⟩) = dom {⟨𝐴, 𝐵⟩}
2 op1st.1 . . 3 𝐴 ∈ V
3 op1st.2 . . 3 𝐵 ∈ V
42, 3op1sta 6075 . 2 dom {⟨𝐴, 𝐵⟩} = 𝐴
51, 4eqtri 2841 1 (1st ‘⟨𝐴, 𝐵⟩) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  wcel 2105  Vcvv 3492  {csn 4557  cop 4563   cuni 4830  dom cdm 5548  cfv 6348  1st c1st 7676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fv 6356  df-1st 7678
This theorem is referenced by:  op1std  7688  op1stg  7690  1stval2  7695  fo1stres  7704  opreuopreu  7723  eloprabi  7750  algrflem  7808  xpmapenlem  8672  fseqenlem2  9439  archnq  10390  ruclem8  15578  idfu1st  17137  cofu1st  17141  xpccatid  17426  prf1st  17442  yonedalem21  17511  yonedalem22  17516  2ndcctbss  21991  upxp  22159  uptx  22161  cnheiborlem  23485  ovollb2lem  24016  ovolctb  24018  ovoliunlem2  24031  ovolshftlem1  24037  ovolscalem1  24041  ovolicc1  24044  addsqnreup  25946  2sqreuop  25965  2sqreuopnn  25966  2sqreuoplt  25967  2sqreuopltb  25968  2sqreuopnnlt  25969  2sqreuopnnltb  25970  ex-1st  28150  cnnvg  28382  cnnvs  28384  h2hva  28678  h2hsm  28679  hhssva  28961  hhsssm  28962  hhshsslem1  28971  eulerpartlemgvv  31533  eulerpartlemgh  31535  satfv0fvfmla0  32557  filnetlem3  33625  poimirlem17  34790  heiborlem8  34977  dvhvaddass  38113  dvhlveclem  38124  diblss  38186  pellexlem5  39308  pellex  39310  dvnprodlem1  42107  hoicvr  42707  hoicvrrex  42715  ovn0lem  42724  ovnhoilem1  42760
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