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Theorem relelrnb 4824
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 4777 . . 3 (𝐴 ∈ ran 𝑅 → (𝐴 ∈ ran 𝑅 ↔ ∃𝑥 𝑥𝑅𝐴))
21ibi 175 . 2 (𝐴 ∈ ran 𝑅 → ∃𝑥 𝑥𝑅𝐴)
3 relelrn 4822 . . . 4 ((Rel 𝑅𝑥𝑅𝐴) → 𝐴 ∈ ran 𝑅)
43ex 114 . . 3 (Rel 𝑅 → (𝑥𝑅𝐴𝐴 ∈ ran 𝑅))
54exlimdv 1799 . 2 (Rel 𝑅 → (∃𝑥 𝑥𝑅𝐴𝐴 ∈ ran 𝑅))
62, 5impbid2 142 1 (Rel 𝑅 → (𝐴 ∈ ran 𝑅 ↔ ∃𝑥 𝑥𝑅𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wex 1472  wcel 2128   class class class wbr 3965  ran crn 4587  Rel wrel 4591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-xp 4592  df-rel 4593  df-cnv 4594  df-dm 4596  df-rn 4597
This theorem is referenced by: (None)
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