MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  op1st Structured version   Unicode version

Theorem op1st 6347
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 6343 . 2  |-  ( 1st `  <. A ,  B >. )  =  U. dom  {
<. A ,  B >. }
2 op1st.1 . . 3  |-  A  e. 
_V
3 op1st.2 . . 3  |-  B  e. 
_V
42, 3op1sta 5343 . 2  |-  U. dom  {
<. A ,  B >. }  =  A
51, 4eqtri 2455 1  |-  ( 1st `  <. A ,  B >. )  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725   _Vcvv 2948   {csn 3806   <.cop 3809   U.cuni 4007   dom cdm 4870   ` cfv 5446   1stc1st 6339
This theorem is referenced by:  op1std  6349  op1stg  6351  1stval2  6356  fo1stres  6362  eloprabi  6405  algrflem  6447  xpmapenlem  7266  fseqenlem2  7898  archnq  8849  ruclem8  12828  idfu1st  14068  cofu1st  14072  xpccatid  14277  prf1st  14293  yonedalem21  14362  yonedalem22  14367  2ndcctbss  17510  upxp  17647  uptx  17649  cnheiborlem  18971  ovollb2lem  19376  ovolctb  19378  ovoliunlem2  19391  ovolshftlem1  19397  ovolscalem1  19401  ovolicc1  19404  ex-1st  21744  cnnvg  22161  cnnvs  22164  h2hva  22469  h2hsm  22470  hhssva  22751  hhsssm  22752  hhshsslem1  22759  br1steq  25390  filnetlem3  26400  heiborlem8  26518  pellexlem5  26887  pellex  26889  dvhvaddass  31832  dvhlveclem  31843  diblss  31905
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fv 5454  df-1st 6341
  Copyright terms: Public domain W3C validator