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Theorem ecelqsdm 6112
Description: Membership of an equivalence class in a quotient set. (Contributed by NM, 30-Jul-1995.)
Assertion
Ref Expression
ecelqsdm ((dom 𝑅 = A [B]𝑅 (A / 𝑅)) → B A)

Proof of Theorem ecelqsdm
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 elqsn0m 6110 . . 3 ((dom 𝑅 = A [B]𝑅 (A / 𝑅)) → x x [B]𝑅)
2 ecdmn0m 6084 . . 3 (B dom 𝑅x x [B]𝑅)
31, 2sylibr 137 . 2 ((dom 𝑅 = A [B]𝑅 (A / 𝑅)) → B dom 𝑅)
4 simpl 102 . 2 ((dom 𝑅 = A [B]𝑅 (A / 𝑅)) → dom 𝑅 = A)
53, 4eleqtrd 2113 1 ((dom 𝑅 = A [B]𝑅 (A / 𝑅)) → B A)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242  wex 1378   wcel 1390  dom cdm 4288  [cec 6040   / cqs 6041
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 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  df-qs 6048
This theorem is referenced by:  th3qlem1  6144  nnnq0lem1  6428  prsrlem1  6650  gt0srpr  6656
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