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Theorem xpexgALT 5904
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 4552 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 3785 . . . 4  |-  U_ y  e.  B  { y }  =  B
21xpeq2i 4459 . . 3  |-  ( A  X.  U_ y  e.  B  { y } )  =  ( A  X.  B )
3 xpiundi 4496 . . 3  |-  ( A  X.  U_ y  e.  B  { y } )  =  U_ y  e.  B  ( A  X.  { y } )
42, 3eqtr3i 2110 . 2  |-  ( A  X.  B )  = 
U_ y  e.  B  ( A  X.  { y } )
5 id 19 . . 3  |-  ( B  e.  W  ->  B  e.  W )
6 fconstmpt 4485 . . . . 5  |-  ( A  X.  { y } )  =  ( x  e.  A  |->  y )
7 mptexg 5522 . . . . 5  |-  ( A  e.  V  ->  (
x  e.  A  |->  y )  e.  _V )
86, 7syl5eqel 2174 . . . 4  |-  ( A  e.  V  ->  ( A  X.  { y } )  e.  _V )
98ralrimivw 2447 . . 3  |-  ( A  e.  V  ->  A. y  e.  B  ( A  X.  { y } )  e.  _V )
10 iunexg 5890 . . 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 284 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  U_ y  e.  B  ( A  X.  { y } )  e.  _V )
124, 11syl5eqel 2174 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  B
)  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    e. wcel 1438   A.wral 2359   _Vcvv 2619   {csn 3446   U_ciun 3730    |-> cmpt 3899    X. cxp 4436
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023
This theorem is referenced by: (None)
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