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Theorem dfrn2 5736
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 5543 . 2 ran 𝐴 = dom 𝐴
2 df-dm 5542 . 2 dom 𝐴 = {𝑦 ∣ ∃𝑥 𝑦𝐴𝑥}
3 vex 3472 . . . . 5 𝑦 ∈ V
4 vex 3472 . . . . 5 𝑥 ∈ V
53, 4brcnv 5730 . . . 4 (𝑦𝐴𝑥𝑥𝐴𝑦)
65exbii 1849 . . 3 (∃𝑥 𝑦𝐴𝑥 ↔ ∃𝑥 𝑥𝐴𝑦)
76abbii 2887 . 2 {𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
81, 2, 73eqtri 2849 1 ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  wex 1781  {cab 2800   class class class wbr 5042  ccnv 5531  dom cdm 5532  ran crn 5533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-opab 5105  df-cnv 5540  df-dm 5542  df-rn 5543
This theorem is referenced by:  dfrn3  5737  dfdm4  5741  dm0rn0  5772  rnep  5774  dfrnf  5797  dfima2  5909  funcnv3  6403  opabrn  30371  rncossdmcoss  35817
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