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Theorem op1st 7218
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 7212 . 2 (1st ‘⟨𝐴, 𝐵⟩) = dom {⟨𝐴, 𝐵⟩}
2 op1st.1 . . 3 𝐴 ∈ V
3 op1st.2 . . 3 𝐵 ∈ V
42, 3op1sta 5654 . 2 dom {⟨𝐴, 𝐵⟩} = 𝐴
51, 4eqtri 2673 1 (1st ‘⟨𝐴, 𝐵⟩) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  wcel 2030  Vcvv 3231  {csn 4210  cop 4216   cuni 4468  dom cdm 5143  cfv 5926  1st c1st 7208
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fv 5934  df-1st 7210
This theorem is referenced by:  op1std  7220  op1stg  7222  1stval2  7227  fo1stres  7236  eloprabi  7277  algrflem  7331  xpmapenlem  8168  fseqenlem2  8886  archnq  9840  ruclem8  15010  idfu1st  16586  cofu1st  16590  xpccatid  16875  prf1st  16891  yonedalem21  16960  yonedalem22  16965  2ndcctbss  21306  upxp  21474  uptx  21476  cnheiborlem  22800  ovollb2lem  23302  ovolctb  23304  ovoliunlem2  23317  ovolshftlem1  23323  ovolscalem1  23327  ovolicc1  23330  wlknwwlksnsur  26844  wlkwwlksur  26851  clwlksfoclwwlk  27050  ex-1st  27431  cnnvg  27661  cnnvs  27663  h2hva  27959  h2hsm  27960  hhssva  28242  hhsssm  28243  hhshsslem1  28252  eulerpartlemgvv  30566  eulerpartlemgh  30568  filnetlem3  32500  poimirlem17  33556  heiborlem8  33747  dvhvaddass  36703  dvhlveclem  36714  diblss  36776  pellexlem5  37714  pellex  37716  dvnprodlem1  40479  hoicvr  41083  hoicvrrex  41091  ovn0lem  41100  ovnhoilem1  41136
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