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Theorem dfrnf 4623
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 4571 . 2  |-  ran  A  =  { w  |  E. v  v A w }
2 nfcv 2223 . . . . 5  |-  F/_ x
v
3 dfrnf.1 . . . . 5  |-  F/_ x A
4 nfcv 2223 . . . . 5  |-  F/_ x w
52, 3, 4nfbr 3849 . . . 4  |-  F/ x  v A w
6 nfv 1462 . . . 4  |-  F/ v  x A w
7 breq1 3808 . . . 4  |-  ( v  =  x  ->  (
v A w  <->  x A w ) )
85, 6, 7cbvex 1681 . . 3  |-  ( E. v  v A w  <->  E. x  x A w )
98abbii 2198 . 2  |-  { w  |  E. v  v A w }  =  {
w  |  E. x  x A w }
10 nfcv 2223 . . . . 5  |-  F/_ y
x
11 dfrnf.2 . . . . 5  |-  F/_ y A
12 nfcv 2223 . . . . 5  |-  F/_ y
w
1310, 11, 12nfbr 3849 . . . 4  |-  F/ y  x A w
1413nfex 1569 . . 3  |-  F/ y E. x  x A w
15 nfv 1462 . . 3  |-  F/ w E. x  x A
y
16 breq2 3809 . . . 4  |-  ( w  =  y  ->  (
x A w  <->  x A
y ) )
1716exbidv 1748 . . 3  |-  ( w  =  y  ->  ( E. x  x A w 
<->  E. x  x A y ) )
1814, 15, 17cbvab 2205 . 2  |-  { w  |  E. x  x A w }  =  {
y  |  E. x  x A y }
191, 9, 183eqtri 2107 1  |-  ran  A  =  { y  |  E. x  x A y }
Colors of variables: wff set class
Syntax hints:    = wceq 1285   E.wex 1422   {cab 2069   F/_wnfc 2210   class class class wbr 3805   ran crn 4392
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-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-cnv 4399  df-dm 4401  df-rn 4402
This theorem is referenced by:  rnopab  4629
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