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Theorem xpexgALT 6229
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 4922 for a version that uses the Power Set axiom instead. (Contributed by Mario Carneiro, 20-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
xpexgALT  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  B
)  e.  _V )

Proof of Theorem xpexgALT
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iunid 4080 . . . 4  |-  U_ y  e.  B  { y }  =  B
21xpeq2i 4832 . . 3  |-  ( A  X.  U_ y  e.  B  { y } )  =  ( A  X.  B )
3 xpiundi 4865 . . 3  |-  ( A  X.  U_ y  e.  B  { y } )  =  U_ y  e.  B  ( A  X.  { y } )
42, 3eqtr3i 2402 . 2  |-  ( A  X.  B )  = 
U_ y  e.  B  ( A  X.  { y } )
5 id 20 . . 3  |-  ( B  e.  W  ->  B  e.  W )
6 fconstmpt 4854 . . . . 5  |-  ( A  X.  { y } )  =  ( x  e.  A  |->  y )
7 mptexg 5897 . . . . 5  |-  ( A  e.  V  ->  (
x  e.  A  |->  y )  e.  _V )
86, 7syl5eqel 2464 . . . 4  |-  ( A  e.  V  ->  ( A  X.  { y } )  e.  _V )
98ralrimivw 2726 . . 3  |-  ( A  e.  V  ->  A. y  e.  B  ( A  X.  { y } )  e.  _V )
10 iunexg 5919 . . 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 465 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  U_ y  e.  B  ( A  X.  { y } )  e.  _V )
124, 11syl5eqel 2464 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  B
)  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1717   A.wral 2642   _Vcvv 2892   {csn 3750   U_ciun 4028    e. cmpt 4200    X. cxp 4809
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395
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