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Theorem elrn2 5397
Description: Membership in a range. (Contributed by NM, 10-Jul-1994.)
Hypothesis
Ref Expression
elrn.1 𝐴 ∈ V
Assertion
Ref Expression
elrn2 (𝐴 ∈ ran 𝐵 ↔ ∃𝑥𝑥, 𝐴⟩ ∈ 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem elrn2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 elrn.1 . 2 𝐴 ∈ V
2 opeq2 4434 . . . 4 (𝑦 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝐴⟩)
32eleq1d 2715 . . 3 (𝑦 = 𝐴 → (⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑥, 𝐴⟩ ∈ 𝐵))
43exbidv 1890 . 2 (𝑦 = 𝐴 → (∃𝑥𝑥, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑥𝑥, 𝐴⟩ ∈ 𝐵))
5 dfrn3 5344 . 2 ran 𝐵 = {𝑦 ∣ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐵}
61, 4, 5elab2 3386 1 (𝐴 ∈ ran 𝐵 ↔ ∃𝑥𝑥, 𝐴⟩ ∈ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1523  wex 1744  wcel 2030  Vcvv 3231  cop 4216  ran crn 5144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-cnv 5151  df-dm 5153  df-rn 5154
This theorem is referenced by:  elrn  5398  dmrnssfld  5416  rniun  5578  ssrnres  5607  relssdmrn  5694  fvelrn  6392  tz7.48-1  7583  prsrn  30089  dfrn5  31801
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