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Theorem releldmb 4514
 Description: Membership in a domain. (Contributed by Mario Carneiro, 5-Nov-2015.)
Assertion
Ref Expression
releldmb (Rel 𝑅 → (A dom 𝑅x A𝑅x))
Distinct variable groups:   x,A   x,𝑅

Proof of Theorem releldmb
StepHypRef Expression
1 eldmg 4473 . . 3 (A dom 𝑅 → (A dom 𝑅x A𝑅x))
21ibi 165 . 2 (A dom 𝑅x A𝑅x)
3 releldm 4512 . . . 4 ((Rel 𝑅 A𝑅x) → A dom 𝑅)
43ex 108 . . 3 (Rel 𝑅 → (A𝑅xA dom 𝑅))
54exlimdv 1697 . 2 (Rel 𝑅 → (x A𝑅xA dom 𝑅))
62, 5impbid2 131 1 (Rel 𝑅 → (A dom 𝑅x A𝑅x))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∃wex 1378   ∈ wcel 1390   class class class wbr 3755  dom cdm 4288  Rel wrel 4293 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-dm 4298 This theorem is referenced by: (None)
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