Theorem 1st2nd2 6178

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 6172	. 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 1353   e. wcel 2148   <.cop 3597   X. cxp 4626   ` cfv 5218   1stc1st 6141   2ndc2nd 6142

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-iota 5180  df-fun 5220  df-fv 5226  df-1st 6143  df-2nd 6144

This theorem is referenced by:  xpopth  6179  eqop  6180  2nd1st  6183  1st2nd  6184  xpmapenlem  6851  djuf1olem  7054  exmidapne  7261  dfplpq2  7355  dfmpq2  7356  enqbreq2  7358  enqdc1  7363  preqlu  7473  prop  7476  elnp1st2nd  7477  cauappcvgprlemladd  7659  elreal2  7831  cnref1o  9652  frecuzrdgrrn  10410  frec2uzrdg  10411  frecuzrdgrcl  10412  frecuzrdgsuc  10416  frecuzrdgrclt  10417  frecuzrdgg  10418  frecuzrdgdomlem  10419  frecuzrdgfunlem  10421  frecuzrdgsuctlem  10425  seq3val  10460  seqvalcd  10461  eucalgval  12056  eucalginv  12058  eucalglt  12059  eucalg  12061  sqpweven  12177  2sqpwodd  12178  qnumdenbi  12194  xpsff1o  12773  tx1cn  13808  tx2cn  13809  txdis  13816  psmetxrge0  13871  xmetxpbl  14047