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Theorem op1st 7136
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 7130 . 2 (1st ‘⟨𝐴, 𝐵⟩) = dom {⟨𝐴, 𝐵⟩}
2 op1st.1 . . 3 𝐴 ∈ V
3 op1st.2 . . 3 𝐵 ∈ V
42, 3op1sta 5586 . 2 dom {⟨𝐴, 𝐵⟩} = 𝐴
51, 4eqtri 2643 1 (1st ‘⟨𝐴, 𝐵⟩) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wcel 1987  Vcvv 3190  {csn 4155  cop 4161   cuni 4409  dom cdm 5084  cfv 5857  1st c1st 7126
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 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
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 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-iota 5820  df-fun 5859  df-fv 5865  df-1st 7128
This theorem is referenced by:  op1std  7138  op1stg  7140  1stval2  7145  fo1stres  7152  eloprabi  7192  algrflem  7246  xpmapenlem  8087  fseqenlem2  8808  archnq  9762  ruclem8  14910  idfu1st  16479  cofu1st  16483  xpccatid  16768  prf1st  16784  yonedalem21  16853  yonedalem22  16858  2ndcctbss  21198  upxp  21366  uptx  21368  cnheiborlem  22693  ovollb2lem  23196  ovolctb  23198  ovoliunlem2  23211  ovolshftlem1  23217  ovolscalem1  23221  ovolicc1  23224  wlknwwlksnsur  26679  wlkwwlksur  26686  clwlksfoclwwlk  26863  ex-1st  27189  cnnvg  27421  cnnvs  27423  h2hva  27719  h2hsm  27720  hhssva  28002  hhsssm  28003  hhshsslem1  28012  eulerpartlemgvv  30261  eulerpartlemgh  30263  br1steq  31427  filnetlem3  32070  poimirlem17  33097  heiborlem8  33288  dvhvaddass  35905  dvhlveclem  35916  diblss  35978  pellexlem5  36916  pellex  36918  dvnprodlem1  39498  hoicvr  40099  hoicvrrex  40107  ovn0lem  40116  ovnhoilem1  40152
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