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Theorem dfrn2 4551
Description: Alternate definition of range. Definition 4 of [Suppes] p. 60. (Contributed by NM, 27-Dec-1996.)
Assertion
Ref Expression
dfrn2 ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem dfrn2
StepHypRef Expression
1 df-rn 4382 . 2 ran 𝐴 = dom 𝐴
2 df-dm 4381 . 2 dom 𝐴 = {𝑦 ∣ ∃𝑥 𝑦𝐴𝑥}
3 vex 2605 . . . . 5 𝑦 ∈ V
4 vex 2605 . . . . 5 𝑥 ∈ V
53, 4brcnv 4546 . . . 4 (𝑦𝐴𝑥𝑥𝐴𝑦)
65exbii 1537 . . 3 (∃𝑥 𝑦𝐴𝑥 ↔ ∃𝑥 𝑥𝐴𝑦)
76abbii 2195 . 2 {𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
81, 2, 73eqtri 2106 1 ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
Colors of variables: wff set class
Syntax hints:   = wceq 1285  wex 1422  {cab 2068   class class class wbr 3793  ccnv 4370  dom cdm 4371  ran crn 4372
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 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-cnv 4379  df-dm 4381  df-rn 4382
This theorem is referenced by:  dfrn3  4552  dfdm4  4555  dm0rn0  4580  dmmrnm  4582  dfrnf  4603  dfima2  4700  funcnv3  4992
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