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Theorem qsel 6094
Description: If an element of a quotient set contains a given element, it is equal to the equivalence class of the element. (Contributed by Mario Carneiro, 12-Aug-2015.)
Assertion
Ref Expression
qsel ((𝑅 Er 𝑋 B (A / 𝑅) 𝐶 B) → B = [𝐶]𝑅)

Proof of Theorem qsel
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 eqid 2022 . . 3 (A / 𝑅) = (A / 𝑅)
2 eleq2 2083 . . . 4 ([x]𝑅 = B → (𝐶 [x]𝑅𝐶 B))
3 eqeq1 2028 . . . 4 ([x]𝑅 = B → ([x]𝑅 = [𝐶]𝑅B = [𝐶]𝑅))
42, 3imbi12d 223 . . 3 ([x]𝑅 = B → ((𝐶 [x]𝑅 → [x]𝑅 = [𝐶]𝑅) ↔ (𝐶 BB = [𝐶]𝑅)))
5 vex 2538 . . . . . 6 x V
6 elecg 6055 . . . . . 6 ((𝐶 [x]𝑅 x V) → (𝐶 [x]𝑅x𝑅𝐶))
75, 6mpan2 403 . . . . 5 (𝐶 [x]𝑅 → (𝐶 [x]𝑅x𝑅𝐶))
87ibi 165 . . . 4 (𝐶 [x]𝑅x𝑅𝐶)
9 simpll 469 . . . . . 6 (((𝑅 Er 𝑋 x A) x𝑅𝐶) → 𝑅 Er 𝑋)
10 simpr 103 . . . . . 6 (((𝑅 Er 𝑋 x A) x𝑅𝐶) → x𝑅𝐶)
119, 10erthi 6063 . . . . 5 (((𝑅 Er 𝑋 x A) x𝑅𝐶) → [x]𝑅 = [𝐶]𝑅)
1211ex 108 . . . 4 ((𝑅 Er 𝑋 x A) → (x𝑅𝐶 → [x]𝑅 = [𝐶]𝑅))
138, 12syl5 28 . . 3 ((𝑅 Er 𝑋 x A) → (𝐶 [x]𝑅 → [x]𝑅 = [𝐶]𝑅))
141, 4, 13ectocld 6083 . 2 ((𝑅 Er 𝑋 B (A / 𝑅)) → (𝐶 BB = [𝐶]𝑅))
15143impia 1087 1 ((𝑅 Er 𝑋 B (A / 𝑅) 𝐶 B) → B = [𝐶]𝑅)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 873   = wceq 1228   wcel 1374  Vcvv 2535   class class class wbr 3738   Er wer 6014  [cec 6015   / cqs 6016
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-er 6017  df-ec 6019  df-qs 6023
This theorem is referenced by: (None)
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