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Theorem dfdmf 5066
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 4890 . 2  |-  dom  A  =  { w  |  E. v  w A v }
2 nfcv 2574 . . . . 5  |-  F/_ y
w
3 dfdmf.2 . . . . 5  |-  F/_ y A
4 nfcv 2574 . . . . 5  |-  F/_ y
v
52, 3, 4nfbr 4258 . . . 4  |-  F/ y  w A v
6 nfv 1630 . . . 4  |-  F/ v  w A y
7 breq2 4218 . . . 4  |-  ( v  =  y  ->  (
w A v  <->  w A
y ) )
85, 6, 7cbvex 1984 . . 3  |-  ( E. v  w A v  <->  E. y  w A
y )
98abbii 2550 . 2  |-  { w  |  E. v  w A v }  =  {
w  |  E. y  w A y }
10 nfcv 2574 . . . . 5  |-  F/_ x w
11 dfdmf.1 . . . . 5  |-  F/_ x A
12 nfcv 2574 . . . . 5  |-  F/_ x
y
1310, 11, 12nfbr 4258 . . . 4  |-  F/ x  w A y
1413nfex 1866 . . 3  |-  F/ x E. y  w A
y
15 nfv 1630 . . 3  |-  F/ w E. y  x A
y
16 breq1 4217 . . . 4  |-  ( w  =  x  ->  (
w A y  <->  x A
y ) )
1716exbidv 1637 . . 3  |-  ( w  =  x  ->  ( E. y  w A
y  <->  E. y  x A y ) )
1814, 15, 17cbvab 2556 . 2  |-  { w  |  E. y  w A y }  =  {
x  |  E. y  x A y }
191, 9, 183eqtri 2462 1  |-  dom  A  =  { x  |  E. y  x A y }
Colors of variables: wff set class
Syntax hints:   E.wex 1551    = wceq 1653   {cab 2424   F/_wnfc 2561   class class class wbr 4214   dom cdm 4880
This theorem is referenced by:  dmopab  5082
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-dm 4890
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