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Theorem ecelqsdm 6083
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 6081 . . 3 ((dom 𝑅 = A [B]𝑅 (A / 𝑅)) → x x [B]𝑅)
2 ecdmn0m 6055 . . 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 2094 1 ((dom 𝑅 = A [B]𝑅 (A / 𝑅)) → B A)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1226  wex 1358   wcel 1370  dom cdm 4268  [cec 6011   / cqs 6012
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 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-sbc 2738  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-br 3735  df-opab 3789  df-xp 4274  df-cnv 4276  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-ec 6015  df-qs 6019
This theorem is referenced by:  th3qlem1  6115  nnnq0lem1  6295  prsrlem1  6486  gt0srpr  6492
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