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Theorem ecelqsg 6615
Description: Membership of an equivalence class in a quotient set. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.)
Assertion
Ref Expression
ecelqsg ((𝑅𝑉𝐵𝐴) → [𝐵]𝑅 ∈ (𝐴 / 𝑅))

Proof of Theorem ecelqsg
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqid 2189 . . 3 [𝐵]𝑅 = [𝐵]𝑅
2 eceq1 6595 . . . . 5 (𝑥 = 𝐵 → [𝑥]𝑅 = [𝐵]𝑅)
32eqeq2d 2201 . . . 4 (𝑥 = 𝐵 → ([𝐵]𝑅 = [𝑥]𝑅 ↔ [𝐵]𝑅 = [𝐵]𝑅))
43rspcev 2856 . . 3 ((𝐵𝐴 ∧ [𝐵]𝑅 = [𝐵]𝑅) → ∃𝑥𝐴 [𝐵]𝑅 = [𝑥]𝑅)
51, 4mpan2 425 . 2 (𝐵𝐴 → ∃𝑥𝐴 [𝐵]𝑅 = [𝑥]𝑅)
6 ecexg 6564 . . . 4 (𝑅𝑉 → [𝐵]𝑅 ∈ V)
7 elqsg 6612 . . . 4 ([𝐵]𝑅 ∈ V → ([𝐵]𝑅 ∈ (𝐴 / 𝑅) ↔ ∃𝑥𝐴 [𝐵]𝑅 = [𝑥]𝑅))
86, 7syl 14 . . 3 (𝑅𝑉 → ([𝐵]𝑅 ∈ (𝐴 / 𝑅) ↔ ∃𝑥𝐴 [𝐵]𝑅 = [𝑥]𝑅))
98biimpar 297 . 2 ((𝑅𝑉 ∧ ∃𝑥𝐴 [𝐵]𝑅 = [𝑥]𝑅) → [𝐵]𝑅 ∈ (𝐴 / 𝑅))
105, 9sylan2 286 1 ((𝑅𝑉𝐵𝐴) → [𝐵]𝑅 ∈ (𝐴 / 𝑅))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  wcel 2160  wrex 2469  Vcvv 2752  [cec 6558   / cqs 6559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-xp 4650  df-cnv 4652  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-ec 6562  df-qs 6566
This theorem is referenced by:  ecelqsi  6616  qliftlem  6640  eroprf  6655  quseccl0g  13187
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