Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfrn2 Structured version   Visualization version   GIF version

Theorem dfrn2 5544
 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 5354 . 2 ran 𝐴 = dom 𝐴
2 df-dm 5353 . 2 dom 𝐴 = {𝑦 ∣ ∃𝑥 𝑦𝐴𝑥}
3 vex 3418 . . . . 5 𝑦 ∈ V
4 vex 3418 . . . . 5 𝑥 ∈ V
53, 4brcnv 5538 . . . 4 (𝑦𝐴𝑥𝑥𝐴𝑦)
65exbii 1949 . . 3 (∃𝑥 𝑦𝐴𝑥 ↔ ∃𝑥 𝑥𝐴𝑦)
76abbii 2945 . 2 {𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
81, 2, 73eqtri 2854 1 ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1658  ∃wex 1880  {cab 2812   class class class wbr 4874  ◡ccnv 5342  dom cdm 5343  ran crn 5344 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pr 5128 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-rab 3127  df-v 3417  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4875  df-opab 4937  df-cnv 5351  df-dm 5353  df-rn 5354 This theorem is referenced by:  dfrn3  5545  dfdm4  5549  dm0rn0  5575  dfrnf  5598  dfima2  5710  funcnv3  6193  opabrn  29972  rncossdmcoss  34754
 Copyright terms: Public domain W3C validator