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Theorem dfrnf 4518
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 xA
dfrnf.2 yA
Assertion
Ref Expression
dfrnf ran A = {yx xAy}
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 4466 . 2 ran A = {wv vAw}
2 nfcv 2175 . . . . 5 xv
3 dfrnf.1 . . . . 5 xA
4 nfcv 2175 . . . . 5 xw
52, 3, 4nfbr 3799 . . . 4 x vAw
6 nfv 1418 . . . 4 v xAw
7 breq1 3758 . . . 4 (v = x → (vAwxAw))
85, 6, 7cbvex 1636 . . 3 (v vAwx xAw)
98abbii 2150 . 2 {wv vAw} = {wx xAw}
10 nfcv 2175 . . . . 5 yx
11 dfrnf.2 . . . . 5 yA
12 nfcv 2175 . . . . 5 yw
1310, 11, 12nfbr 3799 . . . 4 y xAw
1413nfex 1525 . . 3 yx xAw
15 nfv 1418 . . 3 wx xAy
16 breq2 3759 . . . 4 (w = y → (xAwxAy))
1716exbidv 1703 . . 3 (w = y → (x xAwx xAy))
1814, 15, 17cbvab 2157 . 2 {wx xAw} = {yx xAy}
191, 9, 183eqtri 2061 1 ran A = {yx xAy}
Colors of variables: wff set class
Syntax hints:   = wceq 1242  wex 1378  {cab 2023  wnfc 2162   class class class wbr 3755  ran crn 4289
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-cnv 4296  df-dm 4298  df-rn 4299
This theorem is referenced by:  rnopab  4524
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