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Theorem ecdmn0m 6077
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 6040 . . 3 (x [A]𝑅A V)
32exlimiv 1486 . 2 (x x [A]𝑅A V)
4 eldmg 4472 . . 3 (A V → (A dom 𝑅x A𝑅x))
5 vex 2554 . . . . 5 x V
6 elecg 6073 . . . . 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 3754  dom cdm 4287  [cec 6033
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-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3865  ax-pow 3917  ax-pr 3934
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 3352  df-sn 3372  df-pr 3373  df-op 3375  df-br 3755  df-opab 3809  df-xp 4293  df-cnv 4295  df-dm 4297  df-rn 4298  df-res 4299  df-ima 4300  df-ec 6037
This theorem is referenced by:  ereldm  6078  elqsn0m  6103  ecelqsdm  6105
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