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Theorem relelrnb 5282
 Description: Membership in a range. (Contributed by Mario Carneiro, 5-Nov-2015.)
Assertion
Ref Expression
relelrnb (Rel 𝑅 → (𝐴 ∈ ran 𝑅 ↔ ∃𝑥 𝑥𝑅𝐴))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑅

Proof of Theorem relelrnb
StepHypRef Expression
1 elrng 5236 . . 3 (𝐴 ∈ ran 𝑅 → (𝐴 ∈ ran 𝑅 ↔ ∃𝑥 𝑥𝑅𝐴))
21ibi 255 . 2 (𝐴 ∈ ran 𝑅 → ∃𝑥 𝑥𝑅𝐴)
3 relelrn 5280 . . . 4 ((Rel 𝑅𝑥𝑅𝐴) → 𝐴 ∈ ran 𝑅)
43ex 449 . . 3 (Rel 𝑅 → (𝑥𝑅𝐴𝐴 ∈ ran 𝑅))
54exlimdv 1848 . 2 (Rel 𝑅 → (∃𝑥 𝑥𝑅𝐴𝐴 ∈ ran 𝑅))
62, 5impbid2 215 1 (Rel 𝑅 → (𝐴 ∈ ran 𝑅 ↔ ∃𝑥 𝑥𝑅𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∃wex 1695   ∈ wcel 1977   class class class wbr 4583  ran crn 5039  Rel wrel 5043 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049 This theorem is referenced by: (None)
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