Theorem 1st2nd2 6263

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 6257	. 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 1373   e. wcel 2176   <.cop 3636   X. cxp 4674   ` cfv 5272   1stc1st 6226   2ndc2nd 6227

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-un 4481

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-iota 5233  df-fun 5274  df-fv 5280  df-1st 6228  df-2nd 6229

This theorem is referenced by:  xpopth  6264  eqop  6265  2nd1st  6268  1st2nd  6269  xpmapenlem  6948  opabfi  7037  djuf1olem  7157  exmidapne  7374  dfplpq2  7469  dfmpq2  7470  enqbreq2  7472  enqdc1  7477  preqlu  7587  prop  7590  elnp1st2nd  7591  cauappcvgprlemladd  7773  elreal2  7945  cnref1o  9774  frecuzrdgrrn  10555  frec2uzrdg  10556  frecuzrdgrcl  10557  frecuzrdgsuc  10561  frecuzrdgrclt  10562  frecuzrdgg  10563  frecuzrdgdomlem  10564  frecuzrdgfunlem  10566  frecuzrdgsuctlem  10570  seq3val  10607  seqvalcd  10608  eucalgval  12409  eucalginv  12411  eucalglt  12412  eucalg  12414  sqpweven  12530  2sqpwodd  12531  qnumdenbi  12547  xpsff1o  13214  tx1cn  14774  tx2cn  14775  txdis  14782  psmetxrge0  14837  xmetxpbl  15013