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Theorem bnj1400 33190
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj1400.1  |-  ( y  e.  A  ->  A. x  y  e.  A )
Assertion
Ref Expression
bnj1400  |-  dom  U. A  =  U_ x  e.  A  dom  x
Distinct variable groups:    y, A    x, y
Allowed substitution hint:    A( x)

Proof of Theorem bnj1400
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 dmuni 5212 . 2  |-  dom  U. A  =  U_ z  e.  A  dom  z
2 df-iun 4327 . . 3  |-  U_ x  e.  A  dom  x  =  { y  |  E. x  e.  A  y  e.  dom  x }
3 df-iun 4327 . . . 4  |-  U_ z  e.  A  dom  z  =  { y  |  E. z  e.  A  y  e.  dom  z }
4 bnj1400.1 . . . . . . 7  |-  ( y  e.  A  ->  A. x  y  e.  A )
54nfcii 2619 . . . . . 6  |-  F/_ x A
6 nfcv 2629 . . . . . 6  |-  F/_ z A
7 nfv 1683 . . . . . 6  |-  F/ z  y  e.  dom  x
8 nfv 1683 . . . . . 6  |-  F/ x  y  e.  dom  z
9 dmeq 5203 . . . . . . 7  |-  ( x  =  z  ->  dom  x  =  dom  z )
109eleq2d 2537 . . . . . 6  |-  ( x  =  z  ->  (
y  e.  dom  x  <->  y  e.  dom  z ) )
115, 6, 7, 8, 10cbvrexf 3083 . . . . 5  |-  ( E. x  e.  A  y  e.  dom  x  <->  E. z  e.  A  y  e.  dom  z )
1211abbii 2601 . . . 4  |-  { y  |  E. x  e.  A  y  e.  dom  x }  =  {
y  |  E. z  e.  A  y  e.  dom  z }
133, 12eqtr4i 2499 . . 3  |-  U_ z  e.  A  dom  z  =  { y  |  E. x  e.  A  y  e.  dom  x }
142, 13eqtr4i 2499 . 2  |-  U_ x  e.  A  dom  x  = 
U_ z  e.  A  dom  z
151, 14eqtr4i 2499 1  |-  dom  U. A  =  U_ x  e.  A  dom  x
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1377    = wceq 1379    e. wcel 1767   {cab 2452   E.wrex 2815   U.cuni 4245   U_ciun 4325   dom cdm 4999
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-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
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-ral 2819  df-rex 2820  df-rab 2823  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-uni 4246  df-iun 4327  df-br 4448  df-dm 5009
This theorem is referenced by:  bnj1398  33386  bnj1450  33402  bnj1498  33413  bnj1501  33419
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