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Theorem elqsn0m 6465
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 2117 . 2 (𝐴 / 𝑅) = (𝐴 / 𝑅)
2 eleq2 2181 . . 3 ([𝑦]𝑅 = 𝐵 → (𝑥 ∈ [𝑦]𝑅𝑥𝐵))
32exbidv 1781 . 2 ([𝑦]𝑅 = 𝐵 → (∃𝑥 𝑥 ∈ [𝑦]𝑅 ↔ ∃𝑥 𝑥𝐵))
4 eleq2 2181 . . . 4 (dom 𝑅 = 𝐴 → (𝑦 ∈ dom 𝑅𝑦𝐴))
54biimpar 295 . . 3 ((dom 𝑅 = 𝐴𝑦𝐴) → 𝑦 ∈ dom 𝑅)
6 ecdmn0m 6439 . . 3 (𝑦 ∈ dom 𝑅 ↔ ∃𝑥 𝑥 ∈ [𝑦]𝑅)
75, 6sylib 121 . 2 ((dom 𝑅 = 𝐴𝑦𝐴) → ∃𝑥 𝑥 ∈ [𝑦]𝑅)
81, 3, 7ectocld 6463 1 ((dom 𝑅 = 𝐴𝐵 ∈ (𝐴 / 𝑅)) → ∃𝑥 𝑥𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1316  wex 1453  wcel 1465  dom cdm 4509  [cec 6395   / cqs 6396
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515  df-cnv 4517  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-ec 6399  df-qs 6403
This theorem is referenced by:  elqsn0  6466  ecelqsdm  6467
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