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Theorem dfrnf 4708
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 4655 . 2  |-  ran  A  =  { w  |  E. v  v A w }
2 nfcv 2235 . . . . 5  |-  F/_ x
v
3 dfrnf.1 . . . . 5  |-  F/_ x A
4 nfcv 2235 . . . . 5  |-  F/_ x w
52, 3, 4nfbr 3911 . . . 4  |-  F/ x  v A w
6 nfv 1473 . . . 4  |-  F/ v  x A w
7 breq1 3870 . . . 4  |-  ( v  =  x  ->  (
v A w  <->  x A w ) )
85, 6, 7cbvex 1693 . . 3  |-  ( E. v  v A w  <->  E. x  x A w )
98abbii 2210 . 2  |-  { w  |  E. v  v A w }  =  {
w  |  E. x  x A w }
10 nfcv 2235 . . . . 5  |-  F/_ y
x
11 dfrnf.2 . . . . 5  |-  F/_ y A
12 nfcv 2235 . . . . 5  |-  F/_ y
w
1310, 11, 12nfbr 3911 . . . 4  |-  F/ y  x A w
1413nfex 1580 . . 3  |-  F/ y E. x  x A w
15 nfv 1473 . . 3  |-  F/ w E. x  x A
y
16 breq2 3871 . . . 4  |-  ( w  =  y  ->  (
x A w  <->  x A
y ) )
1716exbidv 1760 . . 3  |-  ( w  =  y  ->  ( E. x  x A w 
<->  E. x  x A y ) )
1814, 15, 17cbvab 2217 . 2  |-  { w  |  E. x  x A w }  =  {
y  |  E. x  x A y }
191, 9, 183eqtri 2119 1  |-  ran  A  =  { y  |  E. x  x A y }
Colors of variables: wff set class
Syntax hints:    = wceq 1296   E.wex 1433   {cab 2081   F/_wnfc 2222   class class class wbr 3867   ran crn 4468
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-br 3868  df-opab 3922  df-cnv 4475  df-dm 4477  df-rn 4478
This theorem is referenced by:  rnopab  4714
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