Theorem 1st2nd2 6347

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 6341	. 2 |- ( A e. ( B X. C ) <-> ( A = <. ( 1st ` A ) , ( 2nd ` A ) >. /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) )
2	1	simplbi 274	1 |- ( A e. ( B X. C ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   <.cop 3676   X. cxp 4729   ` cfv 5333   1stc1st 6310   2ndc2nd 6311

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-iota 5293  df-fun 5335  df-fv 5341  df-1st 6312  df-2nd 6313

This theorem is referenced by:  xpopth  6348  eqop  6349  2nd1st  6352  1st2nd  6353  xpmapenlem  7078  opabfi  7175  djuf1olem  7312  exmidapne  7539  dfplpq2  7634  dfmpq2  7635  enqbreq2  7637  enqdc1  7642  preqlu  7752  prop  7755  elnp1st2nd  7756  cauappcvgprlemladd  7938  elreal2  8110  cnref1o  9946  frecuzrdgrrn  10733  frec2uzrdg  10734  frecuzrdgrcl  10735  frecuzrdgsuc  10739  frecuzrdgrclt  10740  frecuzrdgg  10741  frecuzrdgdomlem  10742  frecuzrdgfunlem  10744  frecuzrdgsuctlem  10748  seq3val  10785  seqvalcd  10786  eucalgval  12706  eucalginv  12708  eucalglt  12709  eucalg  12711  sqpweven  12827  2sqpwodd  12828  qnumdenbi  12844  xpsff1o  13512  tx1cn  15080  tx2cn  15081  txdis  15088  psmetxrge0  15143  xmetxpbl  15319