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Theorem dfdmf 5196
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 5009 . 2  |-  dom  A  =  { w  |  E. v  w A v }
2 nfcv 2629 . . . . 5  |-  F/_ y
w
3 dfdmf.2 . . . . 5  |-  F/_ y A
4 nfcv 2629 . . . . 5  |-  F/_ y
v
52, 3, 4nfbr 4491 . . . 4  |-  F/ y  w A v
6 nfv 1683 . . . 4  |-  F/ v  w A y
7 breq2 4451 . . . 4  |-  ( v  =  y  ->  (
w A v  <->  w A
y ) )
85, 6, 7cbvex 1995 . . 3  |-  ( E. v  w A v  <->  E. y  w A
y )
98abbii 2601 . 2  |-  { w  |  E. v  w A v }  =  {
w  |  E. y  w A y }
10 nfcv 2629 . . . . 5  |-  F/_ x w
11 dfdmf.1 . . . . 5  |-  F/_ x A
12 nfcv 2629 . . . . 5  |-  F/_ x
y
1310, 11, 12nfbr 4491 . . . 4  |-  F/ x  w A y
1413nfex 1895 . . 3  |-  F/ x E. y  w A
y
15 nfv 1683 . . 3  |-  F/ w E. y  x A
y
16 breq1 4450 . . . 4  |-  ( w  =  x  ->  (
w A y  <->  x A
y ) )
1716exbidv 1690 . . 3  |-  ( w  =  x  ->  ( E. y  w A
y  <->  E. y  x A y ) )
1814, 15, 17cbvab 2608 . 2  |-  { w  |  E. y  w A y }  =  {
x  |  E. y  x A y }
191, 9, 183eqtri 2500 1  |-  dom  A  =  { x  |  E. y  x A y }
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379   E.wex 1596   {cab 2452   F/_wnfc 2615   class class class wbr 4447   dom cdm 4999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-dm 5009
This theorem is referenced by:  dmopab  5213
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