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Theorem qsel 8359
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 2798 . . 3 (𝐴 / 𝑅) = (𝐴 / 𝑅)
2 eleq2 2878 . . . 4 ([𝑥]𝑅 = 𝐵 → (𝐶 ∈ [𝑥]𝑅𝐶𝐵))
3 eqeq1 2802 . . . 4 ([𝑥]𝑅 = 𝐵 → ([𝑥]𝑅 = [𝐶]𝑅𝐵 = [𝐶]𝑅))
42, 3imbi12d 348 . . 3 ([𝑥]𝑅 = 𝐵 → ((𝐶 ∈ [𝑥]𝑅 → [𝑥]𝑅 = [𝐶]𝑅) ↔ (𝐶𝐵𝐵 = [𝐶]𝑅)))
5 elecg 8315 . . . . . 6 ((𝐶 ∈ [𝑥]𝑅𝑥 ∈ V) → (𝐶 ∈ [𝑥]𝑅𝑥𝑅𝐶))
65elvd 3447 . . . . 5 (𝐶 ∈ [𝑥]𝑅 → (𝐶 ∈ [𝑥]𝑅𝑥𝑅𝐶))
76ibi 270 . . . 4 (𝐶 ∈ [𝑥]𝑅𝑥𝑅𝐶)
8 simpll 766 . . . . . 6 (((𝑅 Er 𝑋𝑥𝐴) ∧ 𝑥𝑅𝐶) → 𝑅 Er 𝑋)
9 simpr 488 . . . . . 6 (((𝑅 Er 𝑋𝑥𝐴) ∧ 𝑥𝑅𝐶) → 𝑥𝑅𝐶)
108, 9erthi 8323 . . . . 5 (((𝑅 Er 𝑋𝑥𝐴) ∧ 𝑥𝑅𝐶) → [𝑥]𝑅 = [𝐶]𝑅)
1110ex 416 . . . 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 8347 . 2 ((𝑅 Er 𝑋𝐵 ∈ (𝐴 / 𝑅)) → (𝐶𝐵𝐵 = [𝐶]𝑅))
14133impia 1114 1 ((𝑅 Er 𝑋𝐵 ∈ (𝐴 / 𝑅) ∧ 𝐶𝐵) → 𝐵 = [𝐶]𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2111  Vcvv 3441   class class class wbr 5030   Er wer 8269  [cec 8270   / cqs 8271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-er 8272  df-ec 8274  df-qs 8278
This theorem is referenced by:  frgpnabllem2  18987  prter3  36178
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