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Theorem ixpint 7557
Description: The intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 3-Feb-2015.)
Assertion
Ref Expression
ixpint  |-  ( B  =/=  (/)  ->  X_ x  e.  A  |^| B  = 
|^|_ y  e.  B  X_ x  e.  A  y )
Distinct variable groups:    x, y, A    x, B, y

Proof of Theorem ixpint
StepHypRef Expression
1 ixpeq2 7544 . . 3  |-  ( A. x  e.  A  |^| B  =  |^|_ y  e.  B  y  ->  X_ x  e.  A  |^| B  = 
X_ x  e.  A  |^|_ y  e.  B  y )
2 intiin 4356 . . . 4  |-  |^| B  =  |^|_ y  e.  B  y
32a1i 11 . . 3  |-  ( x  e.  A  ->  |^| B  =  |^|_ y  e.  B  y )
41, 3mprg 2795 . 2  |-  X_ x  e.  A  |^| B  = 
X_ x  e.  A  |^|_ y  e.  B  y
5 ixpiin 7556 . 2  |-  ( B  =/=  (/)  ->  X_ x  e.  A  |^|_ y  e.  B  y  =  |^|_ y  e.  B  X_ x  e.  A  y )
64, 5syl5eq 2482 1  |-  ( B  =/=  (/)  ->  X_ x  e.  A  |^| B  = 
|^|_ y  e.  B  X_ x  e.  A  y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1870    =/= wne 2625   (/)c0 3767   |^|cint 4258   |^|_ciin 4303   X_cixp 7530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-nul 4556
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-int 4259  df-iin 4305  df-br 4427  df-opab 4485  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-iota 5565  df-fun 5603  df-fn 5604  df-fv 5609  df-ixp 7531
This theorem is referenced by: (None)
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