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Theorem op1st 7995
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 7989 . 2 (1st ‘⟨𝐴, 𝐵⟩) = dom {⟨𝐴, 𝐵⟩}
2 op1st.1 . . 3 𝐴 ∈ V
3 op1st.2 . . 3 𝐵 ∈ V
42, 3op1sta 6223 . 2 dom {⟨𝐴, 𝐵⟩} = 𝐴
51, 4eqtri 2755 1 (1st ‘⟨𝐴, 𝐵⟩) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534  wcel 2099  Vcvv 3469  {csn 4624  cop 4630   cuni 4903  dom cdm 5672  cfv 6542  1st c1st 7985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6494  df-fun 6544  df-fv 6550  df-1st 7987
This theorem is referenced by:  op1std  7997  op1stg  7999  1stval2  8004  fo1stres  8013  opreuopreu  8032  eloprabi  8061  xpmapenlem  9160  fseqenlem2  10040  archnq  10995  ruclem8  16205  idfu1st  17856  cofu1st  17860  xpccatid  18170  prf1st  18186  yonedalem21  18256  yonedalem22  18261  2ndcctbss  23346  upxp  23514  uptx  23516  cnheiborlem  24867  ovollb2lem  25404  ovolctb  25406  ovoliunlem2  25419  ovolshftlem1  25425  ovolscalem1  25429  ovolicc1  25432  addsqnreup  27363  2sqreuop  27382  2sqreuopnn  27383  2sqreuoplt  27384  2sqreuopltb  27385  2sqreuopnnlt  27386  2sqreuopnnltb  27387  precsexlem1  28092  precsexlem4  28095  ex-1st  30241  cnnvg  30475  cnnvs  30477  h2hva  30771  h2hsm  30772  hhssva  31054  hhsssm  31055  hhshsslem1  31064  gsumhashmul  32748  eulerpartlemgvv  33932  eulerpartlemgh  33934  satfv0fvfmla0  34959  filnetlem3  35800  poimirlem17  37045  heiborlem8  37226  dvhvaddass  40507  dvhlveclem  40518  diblss  40580  aks6d1c3  41527  pellexlem5  42175  pellex  42177  dvnprodlem1  45257  hoicvr  45859  hoicvrrex  45867  ovn0lem  45876  ovnhoilem1  45912  thincciso  47978
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