MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ixpint Structured version   Visualization version   Unicode version

Theorem ixpint 7567
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 7554 . . 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 4323 . . . 4  |-  |^| B  =  |^|_ y  e.  B  y
32a1i 11 . . 3  |-  ( x  e.  A  ->  |^| B  =  |^|_ y  e.  B  y )
41, 3mprg 2770 . 2  |-  X_ x  e.  A  |^| B  = 
X_ x  e.  A  |^|_ y  e.  B  y
5 ixpiin 7566 . 2  |-  ( B  =/=  (/)  ->  X_ x  e.  A  |^|_ y  e.  B  y  =  |^|_ y  e.  B  X_ x  e.  A  y )
64, 5syl5eq 2517 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 1452    e. wcel 1904    =/= wne 2641   (/)c0 3722   |^|cint 4226   |^|_ciin 4270   X_cixp 7540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-nul 4527
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-int 4227  df-iin 4272  df-br 4396  df-opab 4455  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5553  df-fun 5591  df-fn 5592  df-fv 5597  df-ixp 7541
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator