Theorem 1st2nd2 6190

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 6184	. 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 1363   e. wcel 2158   <.cop 3607   X. cxp 4636   ` cfv 5228   1stc1st 6153   2ndc2nd 6154

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-sbc 2975  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-iota 5190  df-fun 5230  df-fv 5236  df-1st 6155  df-2nd 6156

This theorem is referenced by:  xpopth  6191  eqop  6192  2nd1st  6195  1st2nd  6196  xpmapenlem  6863  djuf1olem  7066  exmidapne  7273  dfplpq2  7367  dfmpq2  7368  enqbreq2  7370  enqdc1  7375  preqlu  7485  prop  7488  elnp1st2nd  7489  cauappcvgprlemladd  7671  elreal2  7843  cnref1o  9664  frecuzrdgrrn  10422  frec2uzrdg  10423  frecuzrdgrcl  10424  frecuzrdgsuc  10428  frecuzrdgrclt  10429  frecuzrdgg  10430  frecuzrdgdomlem  10431  frecuzrdgfunlem  10433  frecuzrdgsuctlem  10437  seq3val  10472  seqvalcd  10473  eucalgval  12068  eucalginv  12070  eucalglt  12071  eucalg  12073  sqpweven  12189  2sqpwodd  12190  qnumdenbi  12206  xpsff1o  12787  tx1cn  14065  tx2cn  14066  txdis  14073  psmetxrge0  14128  xmetxpbl  14304