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Theorem ixpint 7560
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 7547 . . 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 4353 . . . 4  |-  |^| B  =  |^|_ y  e.  B  y
32a1i 11 . . 3  |-  ( x  e.  A  ->  |^| B  =  |^|_ y  e.  B  y )
41, 3mprg 2785 . 2  |-  X_ x  e.  A  |^| B  = 
X_ x  e.  A  |^|_ y  e.  B  y
5 ixpiin 7559 . 2  |-  ( B  =/=  (/)  ->  X_ x  e.  A  |^|_ y  e.  B  y  =  |^|_ y  e.  B  X_ x  e.  A  y )
64, 5syl5eq 2475 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 1872    =/= wne 2614   (/)c0 3761   |^|cint 4255   |^|_ciin 4300   X_cixp 7533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-nul 4555
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-int 4256  df-iin 4302  df-br 4424  df-opab 4483  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-iota 5565  df-fun 5603  df-fn 5604  df-fv 5609  df-ixp 7534
This theorem is referenced by: (None)
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