Theorem xp2 4121

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 4119	. . 3 |- (x e. (A X. B) <-> (x e. (V X. V) /\ ((1st` x) e. A /\ (2nd` x) e. B)))
2	1	abbi2i 1581	. 2 |- (A X. B) = {x | (x e. (V X. V) /\ ((1st` x) e. A /\ (2nd` x) e. B))}
3		df-rab 1659	. 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 1505	1 |- (A X. B) = {x e. (V X. V) | ((1st` x) e. A /\ (2nd` x) e. B)}

Syntax hints:   /\ wa 223   = wceq 960   e. wcel 962  {cab 1469  {crab 1655  Vcvv 1818   X. cxp 3184  ` cfv 3198  1stc1st 4093  2ndc2nd 4094

This theorem is referenced by:  unielxp 4123

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-9 969  ax-10 970  ax-11 971  ax-12 972  ax-13 973  ax-14 974  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1127  ax-10o 1144  ax-16 1214  ax-11o 1222  ax-ext 1464  ax-sep 2718  ax-nul 2725  ax-pow 2758  ax-pr 2795  ax-un 2882

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 985  df-sb 1176  df-eu 1386  df-mo 1387  df-clab 1470  df-cleq 1475  df-clel 1478  df-ne 1594  df-ral 1656  df-rex 1657  df-rab 1659  df-v 1819  df-dif 2060  df-un 2061  df-in 2062  df-ss 2064  df-nul 2292  df-pw 2414  df-sn 2424  df-pr 2425  df-op 2428  df-uni 2518  df-br 2635  df-opab 2682  df-id 2851  df-xp 3200  df-rel 3201  df-cnv 3202  df-co 3203  df-dm 3204  df-rn 3205  df-res 3206  df-ima 3207  df-fun 3208  df-fv 3214  df-1st 4095  df-2nd 4096

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