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Theorem dfrn2 5845
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 5645 . 2 ran 𝐴 = dom 𝐴
2 df-dm 5644 . 2 dom 𝐴 = {𝑦 ∣ ∃𝑥 𝑦𝐴𝑥}
3 vex 3448 . . . . 5 𝑦 ∈ V
4 vex 3448 . . . . 5 𝑥 ∈ V
53, 4brcnv 5839 . . . 4 (𝑦𝐴𝑥𝑥𝐴𝑦)
65exbii 1851 . . 3 (∃𝑥 𝑦𝐴𝑥 ↔ ∃𝑥 𝑥𝐴𝑦)
76abbii 2803 . 2 {𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
81, 2, 73eqtri 2765 1 ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wex 1782  {cab 2710   class class class wbr 5106  ccnv 5633  dom cdm 5634  ran crn 5635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-cnv 5642  df-dm 5644  df-rn 5645
This theorem is referenced by:  dfrn3  5846  dfdm4  5852  dm0rn0  5881  rnep  5883  dfrnf  5906  dfima2  6016  funcnv3  6572  opabrn  31577  rncossdmcoss  36963
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