Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  ecdmn0m GIF version

Theorem ecdmn0m 6084
 Description: A representative of an inhabited equivalence class belongs to the domain of the equivalence relation. (Contributed by Jim Kingdon, 21-Aug-2019.)
Assertion
Ref Expression
ecdmn0m (A dom 𝑅x x [A]𝑅)
Distinct variable groups:   x,𝑅   x,A

Proof of Theorem ecdmn0m
StepHypRef Expression
1 elex 2560 . 2 (A dom 𝑅A V)
2 ecexr 6047 . . 3 (x [A]𝑅A V)
32exlimiv 1486 . 2 (x x [A]𝑅A V)
4 eldmg 4473 . . 3 (A V → (A dom 𝑅x A𝑅x))
5 vex 2554 . . . . 5 x V
6 elecg 6080 . . . . 5 ((x V A V) → (x [A]𝑅A𝑅x))
75, 6mpan 400 . . . 4 (A V → (x [A]𝑅A𝑅x))
87exbidv 1703 . . 3 (A V → (x x [A]𝑅x A𝑅x))
94, 8bitr4d 180 . 2 (A V → (A dom 𝑅x x [A]𝑅))
101, 3, 9pm5.21nii 619 1 (A dom 𝑅x x [A]𝑅)
 Colors of variables: wff set class Syntax hints:   ↔ wb 98  ∃wex 1378   ∈ wcel 1390  Vcvv 2551   class class class wbr 3755  dom cdm 4288  [cec 6040 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-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  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-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-ec 6044 This theorem is referenced by:  ereldm  6085  elqsn0m  6110  ecelqsdm  6112
 Copyright terms: Public domain W3C validator