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Theorem op1st 6294
Description: Extract the first member of an ordered pair. (Contributed by NM, 5-Oct-2004.)
Hypotheses
Ref Expression
op1st.1  |-  A  e. 
_V
op1st.2  |-  B  e. 
_V
Assertion
Ref Expression
op1st  |-  ( 1st `  <. A ,  B >. )  =  A

Proof of Theorem op1st
StepHypRef Expression
1 1stval 6290 . 2  |-  ( 1st `  <. A ,  B >. )  =  U. dom  {
<. A ,  B >. }
2 op1st.1 . . 3  |-  A  e. 
_V
3 op1st.2 . . 3  |-  B  e. 
_V
42, 3op1sta 5291 . 2  |-  U. dom  {
<. A ,  B >. }  =  A
51, 4eqtri 2407 1  |-  ( 1st `  <. A ,  B >. )  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1717   _Vcvv 2899   {csn 3757   <.cop 3760   U.cuni 3957   dom cdm 4818   ` cfv 5394   1stc1st 6286
This theorem is referenced by:  op1std  6296  op1stg  6298  1stval2  6303  fo1stres  6309  eloprabi  6352  algrflem  6391  xpmapenlem  7210  fseqenlem2  7839  archnq  8790  ruclem8  12763  idfu1st  14003  cofu1st  14007  xpccatid  14212  prf1st  14228  yonedalem21  14297  yonedalem22  14302  2ndcctbss  17439  upxp  17576  uptx  17578  cnheiborlem  18850  ovollb2lem  19251  ovolctb  19253  ovoliunlem2  19266  ovolshftlem1  19272  ovolscalem1  19276  ovolicc1  19279  ex-1st  21600  cnnvg  22017  cnnvs  22020  h2hva  22325  h2hsm  22326  hhssva  22607  hhsssm  22608  hhshsslem1  22615  br1steq  25154  filnetlem3  26100  heiborlem8  26218  pellexlem5  26587  pellex  26589  dvhvaddass  31212  dvhlveclem  31223  diblss  31285
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-iota 5358  df-fun 5396  df-fv 5402  df-1st 6288
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