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Theorem ixpint 7056
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 7043 . . 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 4113 . . . 4  |-  |^| B  =  |^|_ y  e.  B  y
32a1i 11 . . 3  |-  ( x  e.  A  ->  |^| B  =  |^|_ y  e.  B  y )
41, 3mprg 2743 . 2  |-  X_ x  e.  A  |^| B  = 
X_ x  e.  A  |^|_ y  e.  B  y
5 ixpiin 7055 . 2  |-  ( B  =/=  (/)  ->  X_ x  e.  A  |^|_ y  e.  B  y  =  |^|_ y  e.  B  X_ x  e.  A  y )
64, 5syl5eq 2456 1  |-  ( B  =/=  (/)  ->  X_ x  e.  A  |^| B  = 
|^|_ y  e.  B  X_ x  e.  A  y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721    =/= wne 2575   (/)c0 3596   |^|cint 4018   |^|_ciin 4062   X_cixp 7030
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-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-nul 4306
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 2266  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-int 4019  df-iin 4064  df-br 4181  df-opab 4235  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5385  df-fun 5423  df-fn 5424  df-fv 5429  df-ixp 7031
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