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Theorem 1st2nd2 6810
Description: Reconstruction of a member of a Cartesian product in terms of its ordered pair components. (Contributed by NM, 20-Oct-2013.)
Assertion
Ref Expression
1st2nd2  |-  ( A  e.  ( B  X.  C )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )

Proof of Theorem 1st2nd2
StepHypRef Expression
1 elxp6 6805 . 2  |-  ( A  e.  ( B  X.  C )  <->  ( A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  /\  (
( 1st `  A
)  e.  B  /\  ( 2nd `  A )  e.  C ) ) )
21simplbi 458 1  |-  ( A  e.  ( B  X.  C )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   <.cop 4022    X. cxp 4986   ` cfv 5570   1stc1st 6771   2ndc2nd 6772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fv 5578  df-1st 6773  df-2nd 6774
This theorem is referenced by:  1st2ndb  6811  xpopth  6812  eqop  6813  2nd1st  6818  1st2nd  6819  opiota  6832  disjen  7667  xpmapenlem  7677  mapunen  7679  r0weon  8381  enqbreq2  9287  nqereu  9296  lterpq  9337  elreal2  9498  cnref1o  11216  ruclem6  14055  ruclem8  14057  ruclem9  14058  ruclem12  14061  eucalgval  14298  eucalginv  14300  eucalglt  14301  eucalg  14303  qnumdenbi  14364  isstruct2  14728  xpsff1o  15060  comfffval2  15192  comfeq  15197  idfucl  15372  funcpropd  15391  coapm  15552  xpccatid  15659  1stfcl  15668  2ndfcl  15669  1st2ndprf  15677  xpcpropd  15679  evlfcl  15693  hofcl  15730  hofpropd  15738  yonedalem3  15751  gsum2dlem2  17197  gsum2dOLD  17199  mdetunilem9  19292  tx1cn  20279  tx2cn  20280  txdis  20302  txlly  20306  txnlly  20307  txhaus  20317  txkgen  20322  txcon  20359  utop3cls  20923  ucnima  20953  fmucndlem  20963  psmetxrge0  20986  imasdsf1olem  21045  cnheiborlem  21623  caublcls  21916  bcthlem1  21932  bcthlem2  21933  bcthlem4  21935  bcthlem5  21936  ovolfcl  22047  ovolfioo  22048  ovolficc  22049  ovolficcss  22050  ovolfsval  22051  ovolicc2lem1  22097  ovolicc2lem5  22101  ovolfs2  22149  uniiccdif  22156  uniioovol  22157  uniiccvol  22158  uniioombllem2a  22160  uniioombllem2  22161  uniioombllem3a  22162  uniioombllem3  22163  uniioombllem4  22164  uniioombllem5  22165  uniioombllem6  22166  dyadmbl  22178  fsumvma  23689  wlkcpr  24734  isrusgusrgcl  25138  isrgrac  25139  0eusgraiff0rgracl  25146  ofpreima  27737  ofpreima2  27738  fimaproj  28074  1stmbfm  28471  2ndmbfm  28472  sibfof  28549  oddpwdcv  28561  txsconlem  28952  mpst123  29167  mblfinlem1  30294  mblfinlem2  30295  ftc2nc  30342  heiborlem8  30557  dvnprodlem1  31985  etransclem44  32303  uhgrasiz  32785  isfusgracl  32817  isfusgraclALT  32819  bj-elid  35019  dvhgrp  37250  dvhlveclem  37251
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