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Theorem uniixp 6566
Description: The union of an infinite Cartesian product is included in a Cartesian product. (Contributed by NM, 28-Sep-2006.) (Revised by Mario Carneiro, 24-Jun-2015.)
Assertion
Ref Expression
uniixp  |-  U. X_ x  e.  A  B  C_  ( A  X.  U_ x  e.  A  B )
Distinct variable group:    x, A
Allowed substitution hint:    B( x)

Proof of Theorem uniixp
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 ixpf 6565 . . . . 5  |-  ( f  e.  X_ x  e.  A  B  ->  f : A --> U_ x  e.  A  B
)
2 fssxp 5246 . . . . 5  |-  ( f : A --> U_ x  e.  A  B  ->  f 
C_  ( A  X.  U_ x  e.  A  B
) )
31, 2syl 14 . . . 4  |-  ( f  e.  X_ x  e.  A  B  ->  f  C_  ( A  X.  U_ x  e.  A  B ) )
4 selpw 3481 . . . 4  |-  ( f  e.  ~P ( A  X.  U_ x  e.  A  B )  <->  f  C_  ( A  X.  U_ x  e.  A  B )
)
53, 4sylibr 133 . . 3  |-  ( f  e.  X_ x  e.  A  B  ->  f  e.  ~P ( A  X.  U_ x  e.  A  B )
)
65ssriv 3065 . 2  |-  X_ x  e.  A  B  C_  ~P ( A  X.  U_ x  e.  A  B )
7 sspwuni 3861 . 2  |-  ( X_ x  e.  A  B  C_ 
~P ( A  X.  U_ x  e.  A  B
)  <->  U. X_ x  e.  A  B  C_  ( A  X.  U_ x  e.  A  B
) )
86, 7mpbi 144 1  |-  U. X_ x  e.  A  B  C_  ( A  X.  U_ x  e.  A  B )
Colors of variables: wff set class
Syntax hints:    e. wcel 1461    C_ wss 3035   ~Pcpw 3474   U.cuni 3700   U_ciun 3777    X. cxp 4495   -->wf 5075   X_cixp 6543
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-sbc 2877  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-fv 5087  df-ixp 6544
This theorem is referenced by:  ixpexgg  6567
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