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Theorem ecelqsg 6385
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 2095 . . 3 [𝐵]𝑅 = [𝐵]𝑅
2 eceq1 6367 . . . . 5 (𝑥 = 𝐵 → [𝑥]𝑅 = [𝐵]𝑅)
32eqeq2d 2106 . . . 4 (𝑥 = 𝐵 → ([𝐵]𝑅 = [𝑥]𝑅 ↔ [𝐵]𝑅 = [𝐵]𝑅))
43rspcev 2736 . . 3 ((𝐵𝐴 ∧ [𝐵]𝑅 = [𝐵]𝑅) → ∃𝑥𝐴 [𝐵]𝑅 = [𝑥]𝑅)
51, 4mpan2 417 . 2 (𝐵𝐴 → ∃𝑥𝐴 [𝐵]𝑅 = [𝑥]𝑅)
6 ecexg 6336 . . . 4 (𝑅𝑉 → [𝐵]𝑅 ∈ V)
7 elqsg 6382 . . . 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 292 . 2 ((𝑅𝑉 ∧ ∃𝑥𝐴 [𝐵]𝑅 = [𝑥]𝑅) → [𝐵]𝑅 ∈ (𝐴 / 𝑅))
105, 9sylan2 281 1 ((𝑅𝑉𝐵𝐴) → [𝐵]𝑅 ∈ (𝐴 / 𝑅))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1296  wcel 1445  wrex 2371  Vcvv 2633  [cec 6330   / cqs 6331
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-xp 4473  df-cnv 4475  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-ec 6334  df-qs 6338
This theorem is referenced by:  ecelqsi  6386  qliftlem  6410  eroprf  6425
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