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Theorem xpexgALT 6260
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 4952 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 4110 . . . 4  |-  U_ y  e.  B  { y }  =  B
21xpeq2i 4862 . . 3  |-  ( A  X.  U_ y  e.  B  { y } )  =  ( A  X.  B )
3 xpiundi 4895 . . 3  |-  ( A  X.  U_ y  e.  B  { y } )  =  U_ y  e.  B  ( A  X.  { y } )
42, 3eqtr3i 2430 . 2  |-  ( A  X.  B )  = 
U_ y  e.  B  ( A  X.  { y } )
5 id 20 . . 3  |-  ( B  e.  W  ->  B  e.  W )
6 fconstmpt 4884 . . . . 5  |-  ( A  X.  { y } )  =  ( x  e.  A  |->  y )
7 mptexg 5928 . . . . 5  |-  ( A  e.  V  ->  (
x  e.  A  |->  y )  e.  _V )
86, 7syl5eqel 2492 . . . 4  |-  ( A  e.  V  ->  ( A  X.  { y } )  e.  _V )
98ralrimivw 2754 . . 3  |-  ( A  e.  V  ->  A. y  e.  B  ( A  X.  { y } )  e.  _V )
10 iunexg 5950 . . 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 2492 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 1721   A.wral 2670   _Vcvv 2920   {csn 3778   U_ciun 4057    e. cmpt 4230    X. cxp 4839
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pr 4367  ax-un 4664
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425
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