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Theorem xpexgALT 5922
Description: The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. This version is proven using Replacement; see xpexg 4707 for a version that uses the Power Set axiom instead. (Contributed by Mario Carneiro, 20-May-2013.) (Proof modification is discouraged.)
Assertion
Ref Expression
xpexgALT  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  B
)  e.  _V )

Proof of Theorem xpexgALT
StepHypRef Expression
1 iunid 3855 . . . 4  |-  U_ y  e.  B  { y }  =  B
21xpeq2i 4617 . . 3  |-  ( A  X.  U_ y  e.  B  { y } )  =  ( A  X.  B )
3 xpiundi 4650 . . 3  |-  ( A  X.  U_ y  e.  B  { y } )  =  U_ y  e.  B  ( A  X.  { y } )
42, 3eqtr3i 2275 . 2  |-  ( A  X.  B )  = 
U_ y  e.  B  ( A  X.  { y } )
5 id 21 . . 3  |-  ( B  e.  W  ->  B  e.  W )
6 fconstmpt 4639 . . . . 5  |-  ( A  X.  { y } )  =  ( x  e.  A  |->  y )
7 mptexg 5597 . . . . 5  |-  ( A  e.  V  ->  (
x  e.  A  |->  y )  e.  _V )
86, 7syl5eqel 2337 . . . 4  |-  ( A  e.  V  ->  ( A  X.  { y } )  e.  _V )
98ralrimivw 2589 . . 3  |-  ( A  e.  V  ->  A. y  e.  B  ( A  X.  { y } )  e.  _V )
10 iunexg 5619 . . 3  |-  ( ( B  e.  W  /\  A. y  e.  B  ( A  X.  { y } )  e.  _V )  ->  U_ y  e.  B  ( A  X.  { y } )  e.  _V )
115, 9, 10syl2anr 466 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  U_ y  e.  B  ( A  X.  { y } )  e.  _V )
124, 11syl5eqel 2337 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  B
)  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    e. wcel 1621   A.wral 2509   _Vcvv 2727   {csn 3544   U_ciun 3803    e. cmpt 3974    X. cxp 4578
This theorem is referenced by:  hsmexlem2  7937
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608
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