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Theorem bnj1400 28868
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 4888 . 2  |-  dom  U. A  =  U_ z  e.  A  dom  z
2 df-iun 3907 . . 3  |-  U_ x  e.  A  dom  x  =  { y  |  E. x  e.  A  y  e.  dom  x }
3 df-iun 3907 . . . 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 2410 . . . . . 6  |-  F/_ x A
6 nfcv 2419 . . . . . 6  |-  F/_ z A
7 nfv 1605 . . . . . 6  |-  F/ z  y  e.  dom  x
8 nfv 1605 . . . . . 6  |-  F/ x  y  e.  dom  z
9 dmeq 4879 . . . . . . 7  |-  ( x  =  z  ->  dom  x  =  dom  z )
109eleq2d 2350 . . . . . 6  |-  ( x  =  z  ->  (
y  e.  dom  x  <->  y  e.  dom  z ) )
115, 6, 7, 8, 10cbvrexf 2759 . . . . 5  |-  ( E. x  e.  A  y  e.  dom  x  <->  E. z  e.  A  y  e.  dom  z )
1211abbii 2395 . . . 4  |-  { y  |  E. x  e.  A  y  e.  dom  x }  =  {
y  |  E. z  e.  A  y  e.  dom  z }
133, 12eqtr4i 2306 . . 3  |-  U_ z  e.  A  dom  z  =  { y  |  E. x  e.  A  y  e.  dom  x }
142, 13eqtr4i 2306 . 2  |-  U_ x  e.  A  dom  x  = 
U_ z  e.  A  dom  z
151, 14eqtr4i 2306 1  |-  dom  U. A  =  U_ x  e.  A  dom  x
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1527    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544   U.cuni 3827   U_ciun 3905   dom cdm 4689
This theorem is referenced by:  bnj1398  29064  bnj1450  29080  bnj1498  29091  bnj1501  29097
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-dm 4699
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