Theorem xp2 4098

Description: Representation of cross product based on ordered pair component functions.

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 4096	. . 3 |- (x e. (A X. B) <-> (x e. (V X. V) /\ ((1st` x) e. A /\ (2nd` x) e. B)))
2	1	abbi2i 1572	. 2 |- (A X. B) = {x | (x e. (V X. V) /\ ((1st` x) e. A /\ (2nd` x) e. B))}
3		df-rab 1650	. 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	eqtr4 1496	1 |- (A X. B) = {x e. (V X. V) | ((1st` x) e. A /\ (2nd` x) e. B)}

Syntax hints:   /\ wa 223   = wceq 955   e. wcel 957  {cab 1462  {crab 1646  Vcvv 1808   X. cxp 3164  ` cfv 3178  1stc1st 4070  2ndc2nd 4071

This theorem is referenced by:  unielxp 4100

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-rab 1650  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fv 3194  df-1st 4072  df-2nd 4073

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