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Theorem op1st 6314
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 6310 . 2  |-  ( 1st `  <. A ,  B >. )  =  U. dom  {
<. A ,  B >. }
2 op1st.1 . . 3  |-  A  e. 
_V
3 op1st.2 . . 3  |-  B  e. 
_V
42, 3op1sta 5310 . 2  |-  U. dom  {
<. A ,  B >. }  =  A
51, 4eqtri 2424 1  |-  ( 1st `  <. A ,  B >. )  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1721   _Vcvv 2916   {csn 3774   <.cop 3777   U.cuni 3975   dom cdm 4837   ` cfv 5413   1stc1st 6306
This theorem is referenced by:  op1std  6316  op1stg  6318  1stval2  6323  fo1stres  6329  eloprabi  6372  algrflem  6414  xpmapenlem  7233  fseqenlem2  7862  archnq  8813  ruclem8  12791  idfu1st  14031  cofu1st  14035  xpccatid  14240  prf1st  14256  yonedalem21  14325  yonedalem22  14330  2ndcctbss  17471  upxp  17608  uptx  17610  cnheiborlem  18932  ovollb2lem  19337  ovolctb  19339  ovoliunlem2  19352  ovolshftlem1  19358  ovolscalem1  19362  ovolicc1  19365  ex-1st  21705  cnnvg  22122  cnnvs  22125  h2hva  22430  h2hsm  22431  hhssva  22712  hhsssm  22713  hhshsslem1  22720  br1steq  25344  filnetlem3  26299  heiborlem8  26417  pellexlem5  26786  pellex  26788  dvhvaddass  31580  dvhlveclem  31591  diblss  31653
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-iota 5377  df-fun 5415  df-fv 5421  df-1st 6308
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