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Theorem xpundi 4074
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 2795 . . . . . 6 |- (y e. (B u. C) <-> (y e. B \/ y e. C))
21anbi2i 693 . . . . 5 |- ((x e. A /\ y e. (B u. C)) <-> (x e. A /\ (y e. B \/ y e. C)))
3 andi 835 . . . . 5 |- ((x e. A /\ (y e. B \/ y e. C)) <-> ((x e. A /\ y e. B) \/ (x e. A /\ y e. C)))
42, 3bitri 254 . . . 4 |- ((x e. A /\ y e. (B u. C)) <-> ((x e. A /\ y e. B) \/ (x e. A /\ y e. C)))
54opabbii 3432 . . 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 3441 . . 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 1976 . 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 4018 . 2 |- (A X. (B u. C)) = {<.x, y>. | (x e. A /\ y e. (B u. C))}
9 df-xp 4018 . . 3 |- (A X. B) = {<.x, y>. | (x e. A /\ y e. B)}
10 df-xp 4018 . . 3 |- (A X. C) = {<.x, y>. | (x e. A /\ y e. C)}
119, 10uneq12i 2806 . 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 1983 1 |- (A X. (B u. C)) = ((A X. B) u. (A X. C))
Colors of variables: wff set class
Syntax hints:   \/ wo 376   /\ wa 377   = wceq 1449   e. wcel 1451   u. cun 2655  {copab 3427   X. cxp 4002
This theorem is referenced by:  xpun 4078  xp2cda 6389  xpcdaen 6392  unctb 6576  alephadd 6580
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1367  ax-6 1368  ax-7 1369  ax-gen 1370  ax-8 1453  ax-10 1454  ax-11 1455  ax-12 1456  ax-17 1465  ax-9 1480  ax-4 1486  ax-16 1664  ax-ext 1935
This theorem depends on definitions:  df-bi 185  df-or 378  df-an 379  df-ex 1372  df-sb 1626  df-clab 1941  df-cleq 1946  df-clel 1949  df-v 2360  df-un 2662  df-opab 3428  df-xp 4018
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