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Theorem op1st 6355
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 6351 . 2  |-  ( 1st `  <. A ,  B >. )  =  U. dom  {
<. A ,  B >. }
2 op1st.1 . . 3  |-  A  e. 
_V
3 op1st.2 . . 3  |-  B  e. 
_V
42, 3op1sta 5351 . 2  |-  U. dom  {
<. A ,  B >. }  =  A
51, 4eqtri 2456 1  |-  ( 1st `  <. A ,  B >. )  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725   _Vcvv 2956   {csn 3814   <.cop 3817   U.cuni 4015   dom cdm 4878   ` cfv 5454   1stc1st 6347
This theorem is referenced by:  op1std  6357  op1stg  6359  1stval2  6364  fo1stres  6370  eloprabi  6413  algrflem  6455  xpmapenlem  7274  fseqenlem2  7906  archnq  8857  ruclem8  12836  idfu1st  14076  cofu1st  14080  xpccatid  14285  prf1st  14301  yonedalem21  14370  yonedalem22  14375  2ndcctbss  17518  upxp  17655  uptx  17657  cnheiborlem  18979  ovollb2lem  19384  ovolctb  19386  ovoliunlem2  19399  ovolshftlem1  19405  ovolscalem1  19409  ovolicc1  19412  ex-1st  21752  cnnvg  22169  cnnvs  22172  h2hva  22477  h2hsm  22478  hhssva  22759  hhsssm  22760  hhshsslem1  22767  br1steq  25398  filnetlem3  26409  heiborlem8  26527  pellexlem5  26896  pellex  26898  dvhvaddass  31895  dvhlveclem  31906  diblss  31968
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fv 5462  df-1st 6349
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