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Theorem bnj1400 29144
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 5071 . 2  |-  dom  U. A  =  U_ z  e.  A  dom  z
2 df-iun 4087 . . 3  |-  U_ x  e.  A  dom  x  =  { y  |  E. x  e.  A  y  e.  dom  x }
3 df-iun 4087 . . . 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 2562 . . . . . 6  |-  F/_ x A
6 nfcv 2571 . . . . . 6  |-  F/_ z A
7 nfv 1629 . . . . . 6  |-  F/ z  y  e.  dom  x
8 nfv 1629 . . . . . 6  |-  F/ x  y  e.  dom  z
9 dmeq 5062 . . . . . . 7  |-  ( x  =  z  ->  dom  x  =  dom  z )
109eleq2d 2502 . . . . . 6  |-  ( x  =  z  ->  (
y  e.  dom  x  <->  y  e.  dom  z ) )
115, 6, 7, 8, 10cbvrexf 2919 . . . . 5  |-  ( E. x  e.  A  y  e.  dom  x  <->  E. z  e.  A  y  e.  dom  z )
1211abbii 2547 . . . 4  |-  { y  |  E. x  e.  A  y  e.  dom  x }  =  {
y  |  E. z  e.  A  y  e.  dom  z }
133, 12eqtr4i 2458 . . 3  |-  U_ z  e.  A  dom  z  =  { y  |  E. x  e.  A  y  e.  dom  x }
142, 13eqtr4i 2458 . 2  |-  U_ x  e.  A  dom  x  = 
U_ z  e.  A  dom  z
151, 14eqtr4i 2458 1  |-  dom  U. A  =  U_ x  e.  A  dom  x
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1549    = wceq 1652    e. wcel 1725   {cab 2421   E.wrex 2698   U.cuni 4007   U_ciun 4085   dom cdm 4870
This theorem is referenced by:  bnj1398  29340  bnj1450  29356  bnj1498  29367  bnj1501  29373
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-dm 4880
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