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Theorem elqsn0m 6503
Description: An element of a quotient set is inhabited. (Contributed by Jim Kingdon, 21-Aug-2019.)
Assertion
Ref Expression
elqsn0m ((dom 𝑅 = 𝐴𝐵 ∈ (𝐴 / 𝑅)) → ∃𝑥 𝑥𝐵)
Distinct variable groups:   𝑥,𝑅   𝑥,𝐴   𝑥,𝐵

Proof of Theorem elqsn0m
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eqid 2140 . 2 (𝐴 / 𝑅) = (𝐴 / 𝑅)
2 eleq2 2204 . . 3 ([𝑦]𝑅 = 𝐵 → (𝑥 ∈ [𝑦]𝑅𝑥𝐵))
32exbidv 1798 . 2 ([𝑦]𝑅 = 𝐵 → (∃𝑥 𝑥 ∈ [𝑦]𝑅 ↔ ∃𝑥 𝑥𝐵))
4 eleq2 2204 . . . 4 (dom 𝑅 = 𝐴 → (𝑦 ∈ dom 𝑅𝑦𝐴))
54biimpar 295 . . 3 ((dom 𝑅 = 𝐴𝑦𝐴) → 𝑦 ∈ dom 𝑅)
6 ecdmn0m 6477 . . 3 (𝑦 ∈ dom 𝑅 ↔ ∃𝑥 𝑥 ∈ [𝑦]𝑅)
75, 6sylib 121 . 2 ((dom 𝑅 = 𝐴𝑦𝐴) → ∃𝑥 𝑥 ∈ [𝑦]𝑅)
81, 3, 7ectocld 6501 1 ((dom 𝑅 = 𝐴𝐵 ∈ (𝐴 / 𝑅)) → ∃𝑥 𝑥𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1332  wex 1469  wcel 1481  dom cdm 4545  [cec 6433   / cqs 6434
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4052  ax-pow 4104  ax-pr 4137
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2913  df-un 3078  df-in 3080  df-ss 3087  df-pw 3515  df-sn 3536  df-pr 3537  df-op 3539  df-br 3936  df-opab 3996  df-xp 4551  df-cnv 4553  df-dm 4555  df-rn 4556  df-res 4557  df-ima 4558  df-ec 6437  df-qs 6441
This theorem is referenced by:  elqsn0  6504  ecelqsdm  6505
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