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

Theorem uniixp 7077
Description: The union of an infinite Cartesian product is included in a cross 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 7076 . . . . 5  |-  ( f  e.  X_ x  e.  A  B  ->  f : A --> U_ x  e.  A  B
)
2 fssxp 5594 . . . . 5  |-  ( f : A --> U_ x  e.  A  B  ->  f 
C_  ( A  X.  U_ x  e.  A  B
) )
31, 2syl 16 . . . 4  |-  ( f  e.  X_ x  e.  A  B  ->  f  C_  ( A  X.  U_ x  e.  A  B ) )
4 vex 2951 . . . . 5  |-  f  e. 
_V
54elpw 3797 . . . 4  |-  ( f  e.  ~P ( A  X.  U_ x  e.  A  B )  <->  f  C_  ( A  X.  U_ x  e.  A  B )
)
63, 5sylibr 204 . . 3  |-  ( f  e.  X_ x  e.  A  B  ->  f  e.  ~P ( A  X.  U_ x  e.  A  B )
)
76ssriv 3344 . 2  |-  X_ x  e.  A  B  C_  ~P ( A  X.  U_ x  e.  A  B )
8 sspwuni 4168 . 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
) )
97, 8mpbi 200 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 1725    C_ wss 3312   ~Pcpw 3791   U.cuni 4007   U_ciun 4085    X. cxp 4868   -->wf 5442   X_cixp 7055
This theorem is referenced by:  ixpexg  7078
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ixp 7056
  Copyright terms: Public domain W3C validator