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Theorem dfrnf 4917
Description: Definition of range, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Aug-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
dfrnf.1  |-  F/_ x A
dfrnf.2  |-  F/_ y A
Assertion
Ref Expression
dfrnf  |-  ran  A  =  { y  |  E. x  x A y }
Distinct variable group:    x, y
Allowed substitution hints:    A( x, y)

Proof of Theorem dfrnf
Dummy variables  w  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfrn2 4868 . 2  |-  ran  A  =  { w  |  E. v  v A w }
2 nfcv 2419 . . . . 5  |-  F/_ x
v
3 dfrnf.1 . . . . 5  |-  F/_ x A
4 nfcv 2419 . . . . 5  |-  F/_ x w
52, 3, 4nfbr 4067 . . . 4  |-  F/ x  v A w
6 nfv 1605 . . . 4  |-  F/ v  x A w
7 breq1 4026 . . . 4  |-  ( v  =  x  ->  (
v A w  <->  x A w ) )
85, 6, 7cbvex 1925 . . 3  |-  ( E. v  v A w  <->  E. x  x A w )
98abbii 2395 . 2  |-  { w  |  E. v  v A w }  =  {
w  |  E. x  x A w }
10 nfcv 2419 . . . . 5  |-  F/_ y
x
11 dfrnf.2 . . . . 5  |-  F/_ y A
12 nfcv 2419 . . . . 5  |-  F/_ y
w
1310, 11, 12nfbr 4067 . . . 4  |-  F/ y  x A w
1413nfex 1767 . . 3  |-  F/ y E. x  x A w
15 nfv 1605 . . 3  |-  F/ w E. x  x A
y
16 breq2 4027 . . . 4  |-  ( w  =  y  ->  (
x A w  <->  x A
y ) )
1716exbidv 1612 . . 3  |-  ( w  =  y  ->  ( E. x  x A w 
<->  E. x  x A y ) )
1814, 15, 17cbvab 2401 . 2  |-  { w  |  E. x  x A w }  =  {
y  |  E. x  x A y }
191, 9, 183eqtri 2307 1  |-  ran  A  =  { y  |  E. x  x A y }
Colors of variables: wff set class
Syntax hints:   E.wex 1528    = wceq 1623   {cab 2269   F/_wnfc 2406   class class class wbr 4023   ran crn 4690
This theorem is referenced by:  rnopab  4924
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-cnv 4697  df-dm 4699  df-rn 4700
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