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Theorem uniixp 6622
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 6621 . . . . 5  |-  ( f  e.  X_ x  e.  A  B  ->  f : A --> U_ x  e.  A  B
)
2 fssxp 5297 . . . . 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 velpw 3521 . . . 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 3105 . 2  |-  X_ x  e.  A  B  C_  ~P ( A  X.  U_ x  e.  A  B )
7 sspwuni 3904 . 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 1481    C_ wss 3075   ~Pcpw 3514   U.cuni 3743   U_ciun 3820    X. cxp 4544   -->wf 5126   X_cixp 6599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2913  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-fv 5138  df-ixp 6600
This theorem is referenced by:  ixpexgg  6623
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