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Theorem relelrnb 4661
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 4615 . . 3 (𝐴 ∈ ran 𝑅 → (𝐴 ∈ ran 𝑅 ↔ ∃𝑥 𝑥𝑅𝐴))
21ibi 174 . 2 (𝐴 ∈ ran 𝑅 → ∃𝑥 𝑥𝑅𝐴)
3 relelrn 4659 . . . 4 ((Rel 𝑅𝑥𝑅𝐴) → 𝐴 ∈ ran 𝑅)
43ex 113 . . 3 (Rel 𝑅 → (𝑥𝑅𝐴𝐴 ∈ ran 𝑅))
54exlimdv 1747 . 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 1426  wcel 1438   class class class wbr 3837  ran crn 4429  Rel wrel 4433
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-xp 4434  df-rel 4435  df-cnv 4436  df-dm 4438  df-rn 4439
This theorem is referenced by: (None)
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