Theorem 1st2nd2 6081

Description: Reconstruction of a member of a cross 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

Step	Hyp	Ref	Expression
1		elxp6 6075	. 2 |- ( A e. ( B X. C ) <-> ( A = <. ( 1st ` A ) , ( 2nd ` A ) >. /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) )
2	1	simplbi 272	1 |- ( A e. ( B X. C ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1332   e. wcel 1481   <.cop 3535   X. cxp 4545   ` cfv 5131   1stc1st 6044   2ndc2nd 6045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-iota 5096  df-fun 5133  df-fv 5139  df-1st 6046  df-2nd 6047

This theorem is referenced by:  xpopth  6082  eqop  6083  2nd1st  6086  1st2nd  6087  xpmapenlem  6751  djuf1olem  6946  dfplpq2  7186  dfmpq2  7187  enqbreq2  7189  enqdc1  7194  preqlu  7304  prop  7307  elnp1st2nd  7308  cauappcvgprlemladd  7490  elreal2  7662  cnref1o  9469  frecuzrdgrrn  10212  frec2uzrdg  10213  frecuzrdgrcl  10214  frecuzrdgsuc  10218  frecuzrdgrclt  10219  frecuzrdgg  10220  frecuzrdgdomlem  10221  frecuzrdgfunlem  10223  frecuzrdgsuctlem  10227  seq3val  10262  seqvalcd  10263  eucalgval  11771  eucalginv  11773  eucalglt  11774  eucalg  11776  sqpweven  11889  2sqpwodd  11890  qnumdenbi  11906  tx1cn  12477  tx2cn  12478  txdis  12485  psmetxrge0  12540  xmetxpbl  12716