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Theorem ecdmn0m 6059
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 2543 . 2 (A dom 𝑅A V)
2 ecexr 6022 . . 3 (x [A]𝑅A V)
32exlimiv 1471 . 2 (x x [A]𝑅A V)
4 eldmg 4457 . . 3 (A V → (A dom 𝑅x A𝑅x))
5 vex 2538 . . . . 5 x V
6 elecg 6055 . . . . 5 ((x V A V) → (x [A]𝑅A𝑅x))
75, 6mpan 402 . . . 4 (A V → (x [A]𝑅A𝑅x))
87exbidv 1688 . . 3 (A V → (x x [A]𝑅x A𝑅x))
94, 8bitr4d 180 . 2 (A V → (A dom 𝑅x x [A]𝑅))
101, 3, 9pm5.21nii 607 1 (A dom 𝑅x x [A]𝑅)
Colors of variables: wff set class
Syntax hints:  wb 98  wex 1362   wcel 1374  Vcvv 2535   class class class wbr 3738  dom cdm 4272  [cec 6015
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-xp 4278  df-cnv 4280  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-ec 6019
This theorem is referenced by:  ereldm  6060  elqsn0m  6085  ecelqsdm  6087
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