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Theorem dfrnf 5110
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 5061 . 2  |-  ran  A  =  { w  |  E. v  v A w }
2 nfcv 2574 . . . . 5  |-  F/_ x
v
3 dfrnf.1 . . . . 5  |-  F/_ x A
4 nfcv 2574 . . . . 5  |-  F/_ x w
52, 3, 4nfbr 4258 . . . 4  |-  F/ x  v A w
6 nfv 1630 . . . 4  |-  F/ v  x A w
7 breq1 4217 . . . 4  |-  ( v  =  x  ->  (
v A w  <->  x A w ) )
85, 6, 7cbvex 1984 . . 3  |-  ( E. v  v A w  <->  E. x  x A w )
98abbii 2550 . 2  |-  { w  |  E. v  v A w }  =  {
w  |  E. x  x A w }
10 nfcv 2574 . . . . 5  |-  F/_ y
x
11 dfrnf.2 . . . . 5  |-  F/_ y A
12 nfcv 2574 . . . . 5  |-  F/_ y
w
1310, 11, 12nfbr 4258 . . . 4  |-  F/ y  x A w
1413nfex 1866 . . 3  |-  F/ y E. x  x A w
15 nfv 1630 . . 3  |-  F/ w E. x  x A
y
16 breq2 4218 . . . 4  |-  ( w  =  y  ->  (
x A w  <->  x A
y ) )
1716exbidv 1637 . . 3  |-  ( w  =  y  ->  ( E. x  x A w 
<->  E. x  x A y ) )
1814, 15, 17cbvab 2556 . 2  |-  { w  |  E. x  x A w }  =  {
y  |  E. x  x A y }
191, 9, 183eqtri 2462 1  |-  ran  A  =  { y  |  E. x  x A y }
Colors of variables: wff set class
Syntax hints:   E.wex 1551    = wceq 1653   {cab 2424   F/_wnfc 2561   class class class wbr 4214   ran crn 4881
This theorem is referenced by:  rnopab  5117
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-cnv 4888  df-dm 4890  df-rn 4891
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