ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  breldmg Structured version   GIF version

Theorem breldmg 4484
Description: Membership of first of a binary relation in a domain. (Contributed by NM, 21-Mar-2007.)
Assertion
Ref Expression
breldmg ((A 𝐶 B 𝐷 A𝑅B) → A dom 𝑅)

Proof of Theorem breldmg
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 breq2 3759 . . . . 5 (x = B → (A𝑅xA𝑅B))
21spcegv 2635 . . . 4 (B 𝐷 → (A𝑅Bx A𝑅x))
32imp 115 . . 3 ((B 𝐷 A𝑅B) → x A𝑅x)
433adant1 921 . 2 ((A 𝐶 B 𝐷 A𝑅B) → x A𝑅x)
5 eldmg 4473 . . 3 (A 𝐶 → (A dom 𝑅x A𝑅x))
653ad2ant1 924 . 2 ((A 𝐶 B 𝐷 A𝑅B) → (A dom 𝑅x A𝑅x))
74, 6mpbird 156 1 ((A 𝐶 B 𝐷 A𝑅B) → A dom 𝑅)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   w3a 884  wex 1378   wcel 1390   class class class wbr 3755  dom cdm 4288
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-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
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-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-dm 4298
This theorem is referenced by:  brelrng  4508  releldm  4512  brtposg  5810
  Copyright terms: Public domain W3C validator