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Theorem xpundi 4065
Description: Distributive law for cross product over union. Theorem 103 of [Suppes] p. 52.
Assertion
Ref Expression
xpundi |- (A X. (B u. C)) = ((A X. B) u. (A X. C))

Proof of Theorem xpundi
StepHypRef Expression
1 elun 2780 . . . . . 6 |- (y e. (B u. C) <-> (y e. B \/ y e. C))
21anbi2i 679 . . . . 5 |- ((x e. A /\ y e. (B u. C)) <-> (x e. A /\ (y e. B \/ y e. C)))
3 andi 820 . . . . 5 |- ((x e. A /\ (y e. B \/ y e. C)) <-> ((x e. A /\ y e. B) \/ (x e. A /\ y e. C)))
42, 3bitri 239 . . . 4 |- ((x e. A /\ y e. (B u. C)) <-> ((x e. A /\ y e. B) \/ (x e. A /\ y e. C)))
54opabbii 3421 . . 3 |- {<.x, y>. | (x e. A /\ y e. (B u. C))} = {<.x, y>. | ((x e. A /\ y e. B) \/ (x e. A /\ y e. C))}
6 unopab 3430 . . 3 |- ({<.x, y>. | (x e. A /\ y e. B)} u. {<.x, y>. | (x e. A /\ y e. C)}) = {<.x, y>. | ((x e. A /\ y e. B) \/ (x e. A /\ y e. C))}
75, 6eqtr4i 1961 . 2 |- {<.x, y>. | (x e. A /\ y e. (B u. C))} = ({<.x, y>. | (x e. A /\ y e. B)} u. {<.x, y>. | (x e. A /\ y e. C)})
8 df-xp 4009 . 2 |- (A X. (B u. C)) = {<.x, y>. | (x e. A /\ y e. (B u. C))}
9 df-xp 4009 . . 3 |- (A X. B) = {<.x, y>. | (x e. A /\ y e. B)}
10 df-xp 4009 . . 3 |- (A X. C) = {<.x, y>. | (x e. A /\ y e. C)}
119, 10uneq12i 2791 . 2 |- ((A X. B) u. (A X. C)) = ({<.x, y>. | (x e. A /\ y e. B)} u. {<.x, y>. | (x e. A /\ y e. C)})
127, 8, 113eqtr4i 1968 1 |- (A X. (B u. C)) = ((A X. B) u. (A X. C))
Colors of variables: wff set class
Syntax hints:   \/ wo 360   /\ wa 361   = wceq 1434   e. wcel 1436   u. cun 2640  {copab 3416   X. cxp 3993
This theorem is referenced by:  xpun 4069  xp2cda 6372  xpcdaen 6375  unctb 6559  alephadd 6563
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1351  ax-6 1352  ax-7 1353  ax-gen 1354  ax-8 1438  ax-10 1439  ax-11 1440  ax-12 1441  ax-17 1450  ax-9 1465  ax-4 1471  ax-16 1649  ax-ext 1920
This theorem depends on definitions:  df-bi 175  df-or 362  df-an 363  df-ex 1356  df-sb 1611  df-clab 1926  df-cleq 1931  df-clel 1934  df-v 2345  df-un 2647  df-opab 3417  df-xp 4009
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