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Theorem ecelqsdm 6499
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)) → 𝐵𝐴)

Proof of Theorem ecelqsdm
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elqsn0m 6497 . . 3 ((dom 𝑅 = 𝐴 ∧ [𝐵]𝑅 ∈ (𝐴 / 𝑅)) → ∃𝑥 𝑥 ∈ [𝐵]𝑅)
2 ecdmn0m 6471 . . 3 (𝐵 ∈ dom 𝑅 ↔ ∃𝑥 𝑥 ∈ [𝐵]𝑅)
31, 2sylibr 133 . 2 ((dom 𝑅 = 𝐴 ∧ [𝐵]𝑅 ∈ (𝐴 / 𝑅)) → 𝐵 ∈ dom 𝑅)
4 simpl 108 . 2 ((dom 𝑅 = 𝐴 ∧ [𝐵]𝑅 ∈ (𝐴 / 𝑅)) → dom 𝑅 = 𝐴)
53, 4eleqtrd 2218 1 ((dom 𝑅 = 𝐴 ∧ [𝐵]𝑅 ∈ (𝐴 / 𝑅)) → 𝐵𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1331  wex 1468  wcel 1480  dom cdm 4539  [cec 6427   / cqs 6428
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-cnv 4547  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-ec 6431  df-qs 6435
This theorem is referenced by:  th3qlem1  6531  nnnq0lem1  7261  prsrlem1  7557  gt0srpr  7563
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