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Theorem xpundi 4072
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 2800 . . . . . 6 |- (y e. (B u. C) <-> (y e. B \/ y e. C))
21anbi2i 702 . . . . 5 |- ((x e. A /\ y e. (B u. C)) <-> (x e. A /\ (y e. B \/ y e. C)))
3 andi 844 . . . . 5 |- ((x e. A /\ (y e. B \/ y e. C)) <-> ((x e. A /\ y e. B) \/ (x e. A /\ y e. C)))
42, 3bitri 259 . . . 4 |- ((x e. A /\ y e. (B u. C)) <-> ((x e. A /\ y e. B) \/ (x e. A /\ y e. C)))
54opabbii 3437 . . 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 3446 . . 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 1983 . 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 4016 . 2 |- (A X. (B u. C)) = {<.x, y>. | (x e. A /\ y e. (B u. C))}
9 df-xp 4016 . . 3 |- (A X. B) = {<.x, y>. | (x e. A /\ y e. B)}
10 df-xp 4016 . . 3 |- (A X. C) = {<.x, y>. | (x e. A /\ y e. C)}
119, 10uneq12i 2811 . 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 1990 1 |- (A X. (B u. C)) = ((A X. B) u. (A X. C))
Colors of variables: wff set class
Syntax hints:   \/ wo 381   /\ wa 382   = wceq 1457   e. wcel 1459   u. cun 2660  {copab 3432   X. cxp 4000
This theorem is referenced by:  xpun 4076  xp2cda 6357  xpcdaen 6360  unctb 6544  alephadd 6548
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1376  ax-6 1377  ax-7 1378  ax-gen 1379  ax-8 1461  ax-10 1462  ax-11 1463  ax-12 1464  ax-17 1473  ax-9 1488  ax-4 1494  ax-16 1671  ax-ext 1942
This theorem depends on definitions:  df-bi 190  df-or 383  df-an 384  df-ex 1381  df-sb 1633  df-clab 1948  df-cleq 1953  df-clel 1956  df-v 2367  df-un 2667  df-opab 3433  df-xp 4016
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