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Theorem elqsn0m 6204
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 2056 . 2 (𝐴 / 𝑅) = (𝐴 / 𝑅)
2 eleq2 2117 . . 3 ([𝑦]𝑅 = 𝐵 → (𝑥 ∈ [𝑦]𝑅𝑥𝐵))
32exbidv 1722 . 2 ([𝑦]𝑅 = 𝐵 → (∃𝑥 𝑥 ∈ [𝑦]𝑅 ↔ ∃𝑥 𝑥𝐵))
4 eleq2 2117 . . . 4 (dom 𝑅 = 𝐴 → (𝑦 ∈ dom 𝑅𝑦𝐴))
54biimpar 285 . . 3 ((dom 𝑅 = 𝐴𝑦𝐴) → 𝑦 ∈ dom 𝑅)
6 ecdmn0m 6178 . . 3 (𝑦 ∈ dom 𝑅 ↔ ∃𝑥 𝑥 ∈ [𝑦]𝑅)
75, 6sylib 131 . 2 ((dom 𝑅 = 𝐴𝑦𝐴) → ∃𝑥 𝑥 ∈ [𝑦]𝑅)
81, 3, 7ectocld 6202 1 ((dom 𝑅 = 𝐴𝐵 ∈ (𝐴 / 𝑅)) → ∃𝑥 𝑥𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wex 1397  wcel 1409  dom cdm 4372  [cec 6134   / cqs 6135
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2787  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-xp 4378  df-cnv 4380  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-ec 6138  df-qs 6142
This theorem is referenced by:  elqsn0  6205  ecelqsdm  6206
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