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Theorem dfrnf 4886
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 4833 . 2  |-  ran  A  =  { w  |  E. v  v A w }
2 nfcv 2332 . . . . 5  |-  F/_ x
v
3 dfrnf.1 . . . . 5  |-  F/_ x A
4 nfcv 2332 . . . . 5  |-  F/_ x w
52, 3, 4nfbr 4064 . . . 4  |-  F/ x  v A w
6 nfv 1539 . . . 4  |-  F/ v  x A w
7 breq1 4021 . . . 4  |-  ( v  =  x  ->  (
v A w  <->  x A w ) )
85, 6, 7cbvex 1767 . . 3  |-  ( E. v  v A w  <->  E. x  x A w )
98abbii 2305 . 2  |-  { w  |  E. v  v A w }  =  {
w  |  E. x  x A w }
10 nfcv 2332 . . . . 5  |-  F/_ y
x
11 dfrnf.2 . . . . 5  |-  F/_ y A
12 nfcv 2332 . . . . 5  |-  F/_ y
w
1310, 11, 12nfbr 4064 . . . 4  |-  F/ y  x A w
1413nfex 1648 . . 3  |-  F/ y E. x  x A w
15 nfv 1539 . . 3  |-  F/ w E. x  x A
y
16 breq2 4022 . . . 4  |-  ( w  =  y  ->  (
x A w  <->  x A
y ) )
1716exbidv 1836 . . 3  |-  ( w  =  y  ->  ( E. x  x A w 
<->  E. x  x A y ) )
1814, 15, 17cbvab 2313 . 2  |-  { w  |  E. x  x A w }  =  {
y  |  E. x  x A y }
191, 9, 183eqtri 2214 1  |-  ran  A  =  { y  |  E. x  x A y }
Colors of variables: wff set class
Syntax hints:    = wceq 1364   E.wex 1503   {cab 2175   F/_wnfc 2319   class class class wbr 4018   ran crn 4645
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-cnv 4652  df-dm 4654  df-rn 4655
This theorem is referenced by:  rnopab  4892
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