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Theorem op1st 8020
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 8014 . 2 (1st ‘⟨𝐴, 𝐵⟩) = dom {⟨𝐴, 𝐵⟩}
2 op1st.1 . . 3 𝐴 ∈ V
3 op1st.2 . . 3 𝐵 ∈ V
42, 3op1sta 6246 . 2 dom {⟨𝐴, 𝐵⟩} = 𝐴
51, 4eqtri 2762 1 (1st ‘⟨𝐴, 𝐵⟩) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1536  wcel 2105  Vcvv 3477  {csn 4630  cop 4636   cuni 4911  dom cdm 5688  cfv 6562  1st c1st 8010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-iota 6515  df-fun 6564  df-fv 6570  df-1st 8012
This theorem is referenced by:  op1std  8022  op1stg  8024  1stval2  8029  fo1stres  8038  opreuopreu  8057  eloprabi  8086  xpmapenlem  9182  fseqenlem2  10062  archnq  11017  ruclem8  16269  idfu1st  17929  cofu1st  17933  xpccatid  18243  prf1st  18259  yonedalem21  18329  yonedalem22  18334  2ndcctbss  23478  upxp  23646  uptx  23648  cnheiborlem  24999  ovollb2lem  25536  ovolctb  25538  ovoliunlem2  25551  ovolshftlem1  25557  ovolscalem1  25561  ovolicc1  25564  addsqnreup  27501  2sqreuop  27520  2sqreuopnn  27521  2sqreuoplt  27522  2sqreuopltb  27523  2sqreuopnnlt  27524  2sqreuopnnltb  27525  precsexlem1  28245  precsexlem4  28248  ex-1st  30472  cnnvg  30706  cnnvs  30708  h2hva  31002  h2hsm  31003  hhssva  31285  hhsssm  31286  hhshsslem1  31295  gsumhashmul  33046  rlocf1  33259  fracfld  33289  eulerpartlemgvv  34357  eulerpartlemgh  34359  satfv0fvfmla0  35397  filnetlem3  36362  poimirlem17  37623  heiborlem8  37804  dvhvaddass  41079  dvhlveclem  41090  diblss  41152  aks6d1c3  42104  pellexlem5  42820  pellex  42822  dvnprodlem1  45901  hoicvr  46503  hoicvrrex  46511  ovn0lem  46520  ovnhoilem1  46556  gpgedgvtx0  47953  gpgedgvtx1  47954  thincciso  48848
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