Theorem xp2 5925

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 5923	. . 3 |- ( x e. ( A X. B ) <-> ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. A /\ ( 2nd ` x ) e. B ) ) )
2	1	abbi2i 2202	. 2 |- ( A X. B ) = { x | ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. A /\ ( 2nd ` x ) e. B ) ) }
3		df-rab 2368	. 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 2111	1 |- ( A X. B ) = { x e. ( _V X. _V ) | ( ( 1st ` x ) e. A /\ ( 2nd ` x ) e. B ) }

Syntax hints:   /\ wa 102   = wceq 1289   e. wcel 1438   {cab 2074   {crab 2363   _Vcvv 2619   X. cxp 4426   ` cfv 5002   1stc1st 5891   2ndc2nd 5892

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-fo 5008  df-fv 5010  df-1st 5893  df-2nd 5894

This theorem is referenced by:  unielxp  5926