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Theorem relelrnb 5953
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 5898 . . 3 (𝐴 ∈ ran 𝑅 → (𝐴 ∈ ran 𝑅 ↔ ∃𝑥 𝑥𝑅𝐴))
21ibi 266 . 2 (𝐴 ∈ ran 𝑅 → ∃𝑥 𝑥𝑅𝐴)
3 relelrn 5951 . . . 4 ((Rel 𝑅𝑥𝑅𝐴) → 𝐴 ∈ ran 𝑅)
43ex 411 . . 3 (Rel 𝑅 → (𝑥𝑅𝐴𝐴 ∈ ran 𝑅))
54exlimdv 1928 . 2 (Rel 𝑅 → (∃𝑥 𝑥𝑅𝐴𝐴 ∈ ran 𝑅))
62, 5impbid2 225 1 (Rel 𝑅 → (𝐴 ∈ ran 𝑅 ↔ ∃𝑥 𝑥𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wex 1773  wcel 2098   class class class wbr 5152  ran crn 5683  Rel wrel 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-xp 5688  df-rel 5689  df-cnv 5690  df-dm 5692  df-rn 5693
This theorem is referenced by:  iscard4  42994
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