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Theorem dmuni 5046
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 1752 . . . . 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 438 . . . . . . 7  |-  ( ( E. z <. y ,  z >.  e.  x  /\  x  e.  A
)  <->  ( x  e.  A  /\  E. z <. y ,  z >.  e.  x ) )
3 19.41v 1920 . . . . . . 7  |-  ( E. z ( <. y ,  z >.  e.  x  /\  x  e.  A
)  <->  ( E. z <. y ,  z >.  e.  x  /\  x  e.  A ) )
4 vex 2927 . . . . . . . . 9  |-  y  e. 
_V
54eldm2 5035 . . . . . . . 8  |-  ( y  e.  dom  x  <->  E. z <. y ,  z >.  e.  x )
65anbi2i 676 . . . . . . 7  |-  ( ( x  e.  A  /\  y  e.  dom  x )  <-> 
( x  e.  A  /\  E. z <. y ,  z >.  e.  x
) )
72, 3, 63bitr4i 269 . . . . . 6  |-  ( E. z ( <. y ,  z >.  e.  x  /\  x  e.  A
)  <->  ( x  e.  A  /\  y  e. 
dom  x ) )
87exbii 1589 . . . . 5  |-  ( E. x E. z (
<. y ,  z >.  e.  x  /\  x  e.  A )  <->  E. x
( x  e.  A  /\  y  e.  dom  x ) )
91, 8bitri 241 . . . 4  |-  ( E. z E. x (
<. y ,  z >.  e.  x  /\  x  e.  A )  <->  E. x
( x  e.  A  /\  y  e.  dom  x ) )
10 eluni 3986 . . . . 5  |-  ( <.
y ,  z >.  e.  U. A  <->  E. x
( <. y ,  z
>.  e.  x  /\  x  e.  A ) )
1110exbii 1589 . . . 4  |-  ( E. z <. y ,  z
>.  e.  U. A  <->  E. z E. x ( <. y ,  z >.  e.  x  /\  x  e.  A
) )
12 df-rex 2680 . . . 4  |-  ( E. x  e.  A  y  e.  dom  x  <->  E. x
( x  e.  A  /\  y  e.  dom  x ) )
139, 11, 123bitr4i 269 . . 3  |-  ( E. z <. y ,  z
>.  e.  U. A  <->  E. x  e.  A  y  e.  dom  x )
144eldm2 5035 . . 3  |-  ( y  e.  dom  U. A  <->  E. z <. y ,  z
>.  e.  U. A )
15 eliun 4065 . . 3  |-  ( y  e.  U_ x  e.  A  dom  x  <->  E. x  e.  A  y  e.  dom  x )
1613, 14, 153bitr4i 269 . 2  |-  ( y  e.  dom  U. A  <->  y  e.  U_ x  e.  A  dom  x )
1716eqriv 2409 1  |-  dom  U. A  =  U_ x  e.  A  dom  x
Colors of variables: wff set class
Syntax hints:    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721   E.wrex 2675   <.cop 3785   U.cuni 3983   U_ciun 4061   dom cdm 4845
This theorem is referenced by:  tfrlem8  6612  axdc3lem2  8295  wfrlem7  25484  wfrlem9  25486  frrlem5d  25510  frrlem5e  25511  frrlem7  25513  nofulllem5  25582  bnj1400  28925
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-dm 4855
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