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Theorem qsel 8836
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 𝑋𝐵 ∈ (𝐴 / 𝑅) ∧ 𝐶𝐵) → 𝐵 = [𝐶]𝑅)

Proof of Theorem qsel
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . . 3 (𝐴 / 𝑅) = (𝐴 / 𝑅)
2 eleq2 2830 . . . 4 ([𝑥]𝑅 = 𝐵 → (𝐶 ∈ [𝑥]𝑅𝐶𝐵))
3 eqeq1 2741 . . . 4 ([𝑥]𝑅 = 𝐵 → ([𝑥]𝑅 = [𝐶]𝑅𝐵 = [𝐶]𝑅))
42, 3imbi12d 344 . . 3 ([𝑥]𝑅 = 𝐵 → ((𝐶 ∈ [𝑥]𝑅 → [𝑥]𝑅 = [𝐶]𝑅) ↔ (𝐶𝐵𝐵 = [𝐶]𝑅)))
5 elecg 8789 . . . . . 6 ((𝐶 ∈ [𝑥]𝑅𝑥 ∈ V) → (𝐶 ∈ [𝑥]𝑅𝑥𝑅𝐶))
65elvd 3486 . . . . 5 (𝐶 ∈ [𝑥]𝑅 → (𝐶 ∈ [𝑥]𝑅𝑥𝑅𝐶))
76ibi 267 . . . 4 (𝐶 ∈ [𝑥]𝑅𝑥𝑅𝐶)
8 simpll 767 . . . . . 6 (((𝑅 Er 𝑋𝑥𝐴) ∧ 𝑥𝑅𝐶) → 𝑅 Er 𝑋)
9 simpr 484 . . . . . 6 (((𝑅 Er 𝑋𝑥𝐴) ∧ 𝑥𝑅𝐶) → 𝑥𝑅𝐶)
108, 9erthi 8798 . . . . 5 (((𝑅 Er 𝑋𝑥𝐴) ∧ 𝑥𝑅𝐶) → [𝑥]𝑅 = [𝐶]𝑅)
1110ex 412 . . . 4 ((𝑅 Er 𝑋𝑥𝐴) → (𝑥𝑅𝐶 → [𝑥]𝑅 = [𝐶]𝑅))
127, 11syl5 34 . . 3 ((𝑅 Er 𝑋𝑥𝐴) → (𝐶 ∈ [𝑥]𝑅 → [𝑥]𝑅 = [𝐶]𝑅))
131, 4, 12ectocld 8824 . 2 ((𝑅 Er 𝑋𝐵 ∈ (𝐴 / 𝑅)) → (𝐶𝐵𝐵 = [𝐶]𝑅))
14133impia 1118 1 ((𝑅 Er 𝑋𝐵 ∈ (𝐴 / 𝑅) ∧ 𝐶𝐵) → 𝐵 = [𝐶]𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  Vcvv 3480   class class class wbr 5143   Er wer 8742  [cec 8743   / cqs 8744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-er 8745  df-ec 8747  df-qs 8751
This theorem is referenced by:  ghmqusnsg  19300  ghmquskerlem3  19304  ghmqusker  19305  frgpnabllem2  19892  rhmqusnsg  21295  lmhmqusker  33445  rhmquskerlem  33453  prter3  38883
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