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Theorem bnj1400 34295
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 5201 . 2  |-  dom  U. A  =  U_ z  e.  A  dom  z
2 df-iun 4317 . . 3  |-  U_ x  e.  A  dom  x  =  { y  |  E. x  e.  A  y  e.  dom  x }
3 df-iun 4317 . . . 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 2606 . . . . . 6  |-  F/_ x A
6 nfcv 2616 . . . . . 6  |-  F/_ z A
7 nfv 1712 . . . . . 6  |-  F/ z  y  e.  dom  x
8 nfv 1712 . . . . . 6  |-  F/ x  y  e.  dom  z
9 dmeq 5192 . . . . . . 7  |-  ( x  =  z  ->  dom  x  =  dom  z )
109eleq2d 2524 . . . . . 6  |-  ( x  =  z  ->  (
y  e.  dom  x  <->  y  e.  dom  z ) )
115, 6, 7, 8, 10cbvrexf 3076 . . . . 5  |-  ( E. x  e.  A  y  e.  dom  x  <->  E. z  e.  A  y  e.  dom  z )
1211abbii 2588 . . . 4  |-  { y  |  E. x  e.  A  y  e.  dom  x }  =  {
y  |  E. z  e.  A  y  e.  dom  z }
133, 12eqtr4i 2486 . . 3  |-  U_ z  e.  A  dom  z  =  { y  |  E. x  e.  A  y  e.  dom  x }
142, 13eqtr4i 2486 . 2  |-  U_ x  e.  A  dom  x  = 
U_ z  e.  A  dom  z
151, 14eqtr4i 2486 1  |-  dom  U. A  =  U_ x  e.  A  dom  x
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1396    = wceq 1398    e. wcel 1823   {cab 2439   E.wrex 2805   U.cuni 4235   U_ciun 4315   dom cdm 4988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-dm 4998
This theorem is referenced by:  bnj1398  34491  bnj1450  34507  bnj1498  34518  bnj1501  34524
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