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Theorem dfrn3 4551
Description: Alternate definition of range. Definition 6.5(2) of [TakeutiZaring] p. 24. (Contributed by NM, 28-Dec-1996.)
Assertion
Ref Expression
dfrn3 ran 𝐴 = {𝑦 ∣ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴}
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem dfrn3
StepHypRef Expression
1 dfrn2 4550 . 2 ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
2 df-br 3792 . . . 4 (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
32exbii 1512 . . 3 (∃𝑥 𝑥𝐴𝑦 ↔ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴)
43abbii 2169 . 2 {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} = {𝑦 ∣ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴}
51, 4eqtri 2076 1 ran 𝐴 = {𝑦 ∣ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴}
Colors of variables: wff set class
Syntax hints:   = wceq 1259  wex 1397  wcel 1409  {cab 2042  cop 3405   class class class wbr 3791  ran crn 4373
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-cnv 4380  df-dm 4382  df-rn 4383
This theorem is referenced by:  elrn2g  4552  elrn2  4603  imadmrn  4705  imassrn  4706  csbrng  4809
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