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Theorem dfdmf 4556
Description: Definition of domain, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
dfdmf.1  |-  F/_ x A
dfdmf.2  |-  F/_ y A
Assertion
Ref Expression
dfdmf  |-  dom  A  =  { x  |  E. y  x A y }
Distinct variable group:    x, y
Allowed substitution hints:    A( x, y)

Proof of Theorem dfdmf
Dummy variables  w  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-dm 4381 . 2  |-  dom  A  =  { w  |  E. v  w A v }
2 nfcv 2220 . . . . 5  |-  F/_ y
w
3 dfdmf.2 . . . . 5  |-  F/_ y A
4 nfcv 2220 . . . . 5  |-  F/_ y
v
52, 3, 4nfbr 3837 . . . 4  |-  F/ y  w A v
6 nfv 1462 . . . 4  |-  F/ v  w A y
7 breq2 3797 . . . 4  |-  ( v  =  y  ->  (
w A v  <->  w A
y ) )
85, 6, 7cbvex 1680 . . 3  |-  ( E. v  w A v  <->  E. y  w A
y )
98abbii 2195 . 2  |-  { w  |  E. v  w A v }  =  {
w  |  E. y  w A y }
10 nfcv 2220 . . . . 5  |-  F/_ x w
11 dfdmf.1 . . . . 5  |-  F/_ x A
12 nfcv 2220 . . . . 5  |-  F/_ x
y
1310, 11, 12nfbr 3837 . . . 4  |-  F/ x  w A y
1413nfex 1569 . . 3  |-  F/ x E. y  w A
y
15 nfv 1462 . . 3  |-  F/ w E. y  x A
y
16 breq1 3796 . . . 4  |-  ( w  =  x  ->  (
w A y  <->  x A
y ) )
1716exbidv 1747 . . 3  |-  ( w  =  x  ->  ( E. y  w A
y  <->  E. y  x A y ) )
1814, 15, 17cbvab 2202 . 2  |-  { w  |  E. y  w A y }  =  {
x  |  E. y  x A y }
191, 9, 183eqtri 2106 1  |-  dom  A  =  { x  |  E. y  x A y }
Colors of variables: wff set class
Syntax hints:    = wceq 1285   E.wex 1422   {cab 2068   F/_wnfc 2207   class class class wbr 3793   dom cdm 4371
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-dm 4381
This theorem is referenced by:  dmopab  4574
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