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Theorem qsel 6119
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 2037 . . 3 (A / 𝑅) = (A / 𝑅)
2 eleq2 2098 . . . 4 ([x]𝑅 = B → (𝐶 [x]𝑅𝐶 B))
3 eqeq1 2043 . . . 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 2554 . . . . . 6 x V
6 elecg 6080 . . . . . 6 ((𝐶 [x]𝑅 x V) → (𝐶 [x]𝑅x𝑅𝐶))
75, 6mpan2 401 . . . . 5 (𝐶 [x]𝑅 → (𝐶 [x]𝑅x𝑅𝐶))
87ibi 165 . . . 4 (𝐶 [x]𝑅x𝑅𝐶)
9 simpll 481 . . . . . 6 (((𝑅 Er 𝑋 x A) x𝑅𝐶) → 𝑅 Er 𝑋)
10 simpr 103 . . . . . 6 (((𝑅 Er 𝑋 x A) x𝑅𝐶) → x𝑅𝐶)
119, 10erthi 6088 . . . . 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 6108 . 2 ((𝑅 Er 𝑋 B (A / 𝑅)) → (𝐶 BB = [𝐶]𝑅))
15143impia 1100 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 884   = wceq 1242   wcel 1390  Vcvv 2551   class class class wbr 3755   Er wer 6039  [cec 6040   / cqs 6041
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-er 6042  df-ec 6044  df-qs 6048
This theorem is referenced by: (None)
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