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Theorem xpexgALT 6777
Description: Alternate proof of xpexg 6710 requiring Replacement (ax-rep 4558) but not Power Set (ax-pow 4625). (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 4380 . . . 4  |-  U_ y  e.  B  { y }  =  B
21xpeq2i 5020 . . 3  |-  ( A  X.  U_ y  e.  B  { y } )  =  ( A  X.  B )
3 xpiundi 5053 . . 3  |-  ( A  X.  U_ y  e.  B  { y } )  =  U_ y  e.  B  ( A  X.  { y } )
42, 3eqtr3i 2498 . 2  |-  ( A  X.  B )  = 
U_ y  e.  B  ( A  X.  { y } )
5 id 22 . . 3  |-  ( B  e.  W  ->  B  e.  W )
6 fconstmpt 5042 . . . . 5  |-  ( A  X.  { y } )  =  ( x  e.  A  |->  y )
7 mptexg 6129 . . . . 5  |-  ( A  e.  V  ->  (
x  e.  A  |->  y )  e.  _V )
86, 7syl5eqel 2559 . . . 4  |-  ( A  e.  V  ->  ( A  X.  { y } )  e.  _V )
98ralrimivw 2879 . . 3  |-  ( A  e.  V  ->  A. y  e.  B  ( A  X.  { y } )  e.  _V )
10 iunexg 6760 . . 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 478 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  U_ y  e.  B  ( A  X.  { y } )  e.  _V )
124, 11syl5eqel 2559 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  B
)  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1767   A.wral 2814   _Vcvv 3113   {csn 4027   U_ciun 4325    |-> cmpt 4505    X. cxp 4997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595
This theorem is referenced by: (None)
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