Theorem xp2 6228

Description: Representation of cross product based on ordered pair component functions. (Contributed by NM, 16-Sep-2006.)

Assertion

Ref	Expression
xp2	|- ( A X. B ) = { x e. ( _V X. _V ) | ( ( 1st ` x ) e. A /\ ( 2nd ` x ) e. B ) }

Distinct variable groups:   x, A   x, B

Proof of Theorem xp2

Step	Hyp	Ref	Expression
1		elxp7 6225	. . 3 |- ( x e. ( A X. B ) <-> ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. A /\ ( 2nd ` x ) e. B ) ) )
2	1	abbi2i 2308	. 2 |- ( A X. B ) = { x | ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. A /\ ( 2nd ` x ) e. B ) ) }
3		df-rab 2481	. 2 |- { x e. ( _V X. _V ) | ( ( 1st ` x ) e. A /\ ( 2nd ` x ) e. B ) } = { x | ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. A /\ ( 2nd ` x ) e. B ) ) }
4	2, 3	eqtr4i 2217	1 |- ( A X. B ) = { x e. ( _V X. _V ) | ( ( 1st ` x ) e. A /\ ( 2nd ` x ) e. B ) }

Syntax hints:   /\ wa 104   = wceq 1364   e. wcel 2164   {cab 2179   {crab 2476   _Vcvv 2760   X. cxp 4658   ` cfv 5255   1stc1st 6193   2ndc2nd 6194

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-1st 6195  df-2nd 6196

This theorem is referenced by:  unielxp  6229