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Theorem xpundi 4207
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 2938 . . . . . 6 |- (y e. (B u. C) <-> (y e. B \/ y e. C))
21anbi2i 785 . . . . 5 |- ((x e. A /\ y e. (B u. C)) <-> (x e. A /\ (y e. B \/ y e. C)))
3 andi 931 . . . . 5 |- ((x e. A /\ (y e. B \/ y e. C)) <-> ((x e. A /\ y e. B) \/ (x e. A /\ y e. C)))
42, 3bitri 286 . . . 4 |- ((x e. A /\ y e. (B u. C)) <-> ((x e. A /\ y e. B) \/ (x e. A /\ y e. C)))
54opabbii 3572 . . 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 3581 . . 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 2115 . 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 4151 . 2 |- (A X. (B u. C)) = {<.x, y>. | (x e. A /\ y e. (B u. C))}
9 df-xp 4151 . . 3 |- (A X. B) = {<.x, y>. | (x e. A /\ y e. B)}
10 df-xp 4151 . . 3 |- (A X. C) = {<.x, y>. | (x e. A /\ y e. C)}
119, 10uneq12i 2949 . 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 2122 1 |- (A X. (B u. C)) = ((A X. B) u. (A X. C))
Colors of variables: wff set class
Syntax hints:   \/ wo 417   /\ wa 418   = wceq 1592   e. wcel 1594   u. cun 2799  {copab 3567   X. cxp 4135
This theorem is referenced by:  xpun 4209  xp2cda 6498  xpcdaen 6501  unctb 6685  alephadd 6689  phtpycolem5 16690
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1516  ax-6 1517  ax-7 1518  ax-gen 1519  ax-8 1596  ax-10 1597  ax-11 1598  ax-12 1599  ax-17 1608  ax-9 1620  ax-4 1626  ax-16 1803  ax-ext 2074
This theorem depends on definitions:  df-bi 210  df-or 419  df-an 420  df-ex 1521  df-sb 1765  df-clab 2080  df-cleq 2085  df-clel 2088  df-v 2501  df-un 2806  df-opab 3568  df-xp 4151
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