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Theorem qsel 7786
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 2621 . . 3 (𝐴 / 𝑅) = (𝐴 / 𝑅)
2 eleq2 2687 . . . 4 ([𝑥]𝑅 = 𝐵 → (𝐶 ∈ [𝑥]𝑅𝐶𝐵))
3 eqeq1 2625 . . . 4 ([𝑥]𝑅 = 𝐵 → ([𝑥]𝑅 = [𝐶]𝑅𝐵 = [𝐶]𝑅))
42, 3imbi12d 334 . . 3 ([𝑥]𝑅 = 𝐵 → ((𝐶 ∈ [𝑥]𝑅 → [𝑥]𝑅 = [𝐶]𝑅) ↔ (𝐶𝐵𝐵 = [𝐶]𝑅)))
5 vex 3193 . . . . . 6 𝑥 ∈ V
6 elecg 7745 . . . . . 6 ((𝐶 ∈ [𝑥]𝑅𝑥 ∈ V) → (𝐶 ∈ [𝑥]𝑅𝑥𝑅𝐶))
75, 6mpan2 706 . . . . 5 (𝐶 ∈ [𝑥]𝑅 → (𝐶 ∈ [𝑥]𝑅𝑥𝑅𝐶))
87ibi 256 . . . 4 (𝐶 ∈ [𝑥]𝑅𝑥𝑅𝐶)
9 simpll 789 . . . . . 6 (((𝑅 Er 𝑋𝑥𝐴) ∧ 𝑥𝑅𝐶) → 𝑅 Er 𝑋)
10 simpr 477 . . . . . 6 (((𝑅 Er 𝑋𝑥𝐴) ∧ 𝑥𝑅𝐶) → 𝑥𝑅𝐶)
119, 10erthi 7753 . . . . 5 (((𝑅 Er 𝑋𝑥𝐴) ∧ 𝑥𝑅𝐶) → [𝑥]𝑅 = [𝐶]𝑅)
1211ex 450 . . . 4 ((𝑅 Er 𝑋𝑥𝐴) → (𝑥𝑅𝐶 → [𝑥]𝑅 = [𝐶]𝑅))
138, 12syl5 34 . . 3 ((𝑅 Er 𝑋𝑥𝐴) → (𝐶 ∈ [𝑥]𝑅 → [𝑥]𝑅 = [𝐶]𝑅))
141, 4, 13ectocld 7774 . 2 ((𝑅 Er 𝑋𝐵 ∈ (𝐴 / 𝑅)) → (𝐶𝐵𝐵 = [𝐶]𝑅))
15143impia 1258 1 ((𝑅 Er 𝑋𝐵 ∈ (𝐴 / 𝑅) ∧ 𝐶𝐵) → 𝐵 = [𝐶]𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  Vcvv 3190   class class class wbr 4623   Er wer 7699  [cec 7700   / cqs 7701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-er 7702  df-ec 7704  df-qs 7708
This theorem is referenced by:  frgpnabllem2  18217  prter3  33686
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