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Theorem dmuni 4890
Description: The domain of a union. Part of Exercise 8 of [Enderton] p. 41. (Contributed by NM, 3-Feb-2004.)
Assertion
Ref Expression
dmuni  |-  dom  U. A  =  U_ x  e.  A  dom  x
Distinct variable group:    x, A

Proof of Theorem dmuni
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 excom 1788 . . . . 5  |-  ( E. z E. x (
<. y ,  z >.  e.  x  /\  x  e.  A )  <->  E. x E. z ( <. y ,  z >.  e.  x  /\  x  e.  A
) )
2 ancom 437 . . . . . . 7  |-  ( ( E. z <. y ,  z >.  e.  x  /\  x  e.  A
)  <->  ( x  e.  A  /\  E. z <. y ,  z >.  e.  x ) )
3 19.41v 1844 . . . . . . 7  |-  ( E. z ( <. y ,  z >.  e.  x  /\  x  e.  A
)  <->  ( E. z <. y ,  z >.  e.  x  /\  x  e.  A ) )
4 vex 2793 . . . . . . . . 9  |-  y  e. 
_V
54eldm2 4879 . . . . . . . 8  |-  ( y  e.  dom  x  <->  E. z <. y ,  z >.  e.  x )
65anbi2i 675 . . . . . . 7  |-  ( ( x  e.  A  /\  y  e.  dom  x )  <-> 
( x  e.  A  /\  E. z <. y ,  z >.  e.  x
) )
72, 3, 63bitr4i 268 . . . . . 6  |-  ( E. z ( <. y ,  z >.  e.  x  /\  x  e.  A
)  <->  ( x  e.  A  /\  y  e. 
dom  x ) )
87exbii 1571 . . . . 5  |-  ( E. x E. z (
<. y ,  z >.  e.  x  /\  x  e.  A )  <->  E. x
( x  e.  A  /\  y  e.  dom  x ) )
91, 8bitri 240 . . . 4  |-  ( E. z E. x (
<. y ,  z >.  e.  x  /\  x  e.  A )  <->  E. x
( x  e.  A  /\  y  e.  dom  x ) )
10 eluni 3832 . . . . 5  |-  ( <.
y ,  z >.  e.  U. A  <->  E. x
( <. y ,  z
>.  e.  x  /\  x  e.  A ) )
1110exbii 1571 . . . 4  |-  ( E. z <. y ,  z
>.  e.  U. A  <->  E. z E. x ( <. y ,  z >.  e.  x  /\  x  e.  A
) )
12 df-rex 2551 . . . 4  |-  ( E. x  e.  A  y  e.  dom  x  <->  E. x
( x  e.  A  /\  y  e.  dom  x ) )
139, 11, 123bitr4i 268 . . 3  |-  ( E. z <. y ,  z
>.  e.  U. A  <->  E. x  e.  A  y  e.  dom  x )
144eldm2 4879 . . 3  |-  ( y  e.  dom  U. A  <->  E. z <. y ,  z
>.  e.  U. A )
15 eliun 3911 . . 3  |-  ( y  e.  U_ x  e.  A  dom  x  <->  E. x  e.  A  y  e.  dom  x )
1613, 14, 153bitr4i 268 . 2  |-  ( y  e.  dom  U. A  <->  y  e.  U_ x  e.  A  dom  x )
1716eqriv 2282 1  |-  dom  U. A  =  U_ x  e.  A  dom  x
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1530    = wceq 1625    e. wcel 1686   E.wrex 2546   <.cop 3645   U.cuni 3829   U_ciun 3907   dom cdm 4691
This theorem is referenced by:  tfrlem8  6402  axdc3lem2  8079  wfrlem7  24264  wfrlem9  24266  frrlem5d  24290  frrlem5e  24291  frrlem7  24293  nofulllem5  24362  bnj1400  28941
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-dm 4701
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