Theorem 1st2nd2 6409

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 6403	. 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 ) >. )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   <.cop 3712   X. cxp 4772   ` cfv 5377   1stc1st 6372   2ndc2nd 6373

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-iota 5337  df-fun 5379  df-fv 5385  df-1st 6374  df-2nd 6375

This theorem is used by:  xpopth  6410  eqop  6411  2nd1st  6414  1st2nd  6415  xpmapenlem  7149  mapunen  7151  opabfi  7247  djuf1olem  7394  exmidapne  7627  dfplpq2  7722  dfmpq2  7723  enqbreq2  7725  enqdc1  7730  preqlu  7840  prop  7843  elnp1st2nd  7844  cauappcvgprlemladd  8026  elreal2  8198  cnref1o  10062  frecuzrdgrrn  10860  frec2uzrdg  10861  frecuzrdgrcl  10862  frecuzrdgsuc  10866  frecuzrdgrclt  10867  frecuzrdgg  10868  frecuzrdgdomlem  10869  frecuzrdgfunlem  10871  frecuzrdgsuctlem  10875  seq3val  10912  seqvalcd  10913  eucalgval  12851  eucalginv  12853  eucalglt  12854  eucalg  12856  sqpweven  12974  2sqpwodd  12975  qnumdenbi  12991  xpsff1o  13723  tx1cn  15461  tx2cn  15462  txdis  15469  psmetxrge0  15524  xmetxpbl  15700