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Theorem relelrnb 4594
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 4548 . . 3 (𝐴 ∈ ran 𝑅 → (𝐴 ∈ ran 𝑅 ↔ ∃𝑥 𝑥𝑅𝐴))
21ibi 174 . 2 (𝐴 ∈ ran 𝑅 → ∃𝑥 𝑥𝑅𝐴)
3 relelrn 4592 . . . 4 ((Rel 𝑅𝑥𝑅𝐴) → 𝐴 ∈ ran 𝑅)
43ex 113 . . 3 (Rel 𝑅 → (𝑥𝑅𝐴𝐴 ∈ ran 𝑅))
54exlimdv 1741 . 2 (Rel 𝑅 → (∃𝑥 𝑥𝑅𝐴𝐴 ∈ ran 𝑅))
62, 5impbid2 141 1 (Rel 𝑅 → (𝐴 ∈ ran 𝑅 ↔ ∃𝑥 𝑥𝑅𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wex 1422  wcel 1434   class class class wbr 3787  ran crn 4366  Rel wrel 4370
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-br 3788  df-opab 3842  df-xp 4371  df-rel 4372  df-cnv 4373  df-dm 4375  df-rn 4376
This theorem is referenced by: (None)
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