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Theorem ecelqsg 6059
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 ((𝑅 𝑉 B A) → [B]𝑅 (A / 𝑅))

Proof of Theorem ecelqsg
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 eqid 2014 . . 3 [B]𝑅 = [B]𝑅
2 eceq1 6041 . . . . 5 (x = B → [x]𝑅 = [B]𝑅)
32eqeq2d 2025 . . . 4 (x = B → ([B]𝑅 = [x]𝑅 ↔ [B]𝑅 = [B]𝑅))
43rspcev 2625 . . 3 ((B A [B]𝑅 = [B]𝑅) → x A [B]𝑅 = [x]𝑅)
51, 4mpan2 401 . 2 (B Ax A [B]𝑅 = [x]𝑅)
6 ecexg 6010 . . . 4 (𝑅 𝑉 → [B]𝑅 V)
7 elqsg 6056 . . . 4 ([B]𝑅 V → ([B]𝑅 (A / 𝑅) ↔ x A [B]𝑅 = [x]𝑅))
86, 7syl 14 . . 3 (𝑅 𝑉 → ([B]𝑅 (A / 𝑅) ↔ x A [B]𝑅 = [x]𝑅))
98biimpar 281 . 2 ((𝑅 𝑉 x A [B]𝑅 = [x]𝑅) → [B]𝑅 (A / 𝑅))
105, 9sylan2 270 1 ((𝑅 𝑉 B A) → [B]𝑅 (A / 𝑅))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1224   wcel 1367  wrex 2277  Vcvv 2527  [cec 6004   / cqs 6005
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 614  ax-5 1310  ax-7 1311  ax-gen 1312  ax-ie1 1356  ax-ie2 1357  ax-8 1369  ax-10 1370  ax-11 1371  ax-i12 1372  ax-bnd 1373  ax-4 1374  ax-13 1378  ax-14 1379  ax-17 1393  ax-i9 1397  ax-ial 1401  ax-i5r 1402  ax-ext 1996  ax-sep 3839  ax-pow 3891  ax-pr 3908  ax-un 4109
This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1227  df-nf 1324  df-sb 1620  df-eu 1877  df-mo 1878  df-clab 2001  df-cleq 2007  df-clel 2010  df-nfc 2141  df-ral 2281  df-rex 2282  df-v 2529  df-un 2891  df-in 2893  df-ss 2900  df-pw 3326  df-sn 3346  df-pr 3347  df-op 3349  df-uni 3545  df-br 3729  df-opab 3783  df-xp 4267  df-cnv 4269  df-dm 4271  df-rn 4272  df-res 4273  df-ima 4274  df-ec 6008  df-qs 6012
This theorem is referenced by:  ecelqsi  6060  qliftlem  6084  eroprf  6099
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