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Theorem iunxpconst 5055
Description: Membership in a union of Cartesian products when the second factor is constant. (Contributed by Mario Carneiro, 29-Dec-2014.)
Assertion
Ref Expression
iunxpconst  |-  U_ x  e.  A  ( {
x }  X.  B
)  =  ( A  X.  B )
Distinct variable groups:    x, A    x, B

Proof of Theorem iunxpconst
StepHypRef Expression
1 xpiundir 5054 . 2  |-  ( U_ x  e.  A  {
x }  X.  B
)  =  U_ x  e.  A  ( {
x }  X.  B
)
2 iunid 4380 . . 3  |-  U_ x  e.  A  { x }  =  A
32xpeq1i 5019 . 2  |-  ( U_ x  e.  A  {
x }  X.  B
)  =  ( A  X.  B )
41, 3eqtr3i 2498 1  |-  U_ x  e.  A  ( {
x }  X.  B
)  =  ( A  X.  B )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379   {csn 4027   U_ciun 4325    X. cxp 4997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-iun 4327  df-opab 4506  df-xp 5005
This theorem is referenced by:  ralxp  5142  rexxp  5143  mpt2mpt  6376  mpt2mpts  6845  fmpt2  6848  fsumxp  13546  dvfval  22036  indval2  27668  fprodxp  28689  filnetlem3  29801
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