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Theorem eqvrelqsel 37291
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.) (Revised by Peter Mazsa, 28-Dec-2019.)
Assertion
Ref Expression
eqvrelqsel (( EqvRel 𝑅𝐵 ∈ (𝐴 / 𝑅) ∧ 𝐶𝐵) → 𝐵 = [𝐶]𝑅)

Proof of Theorem eqvrelqsel
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqid 2731 . . 3 (𝐴 / 𝑅) = (𝐴 / 𝑅)
2 eleq2 2821 . . . 4 ([𝑥]𝑅 = 𝐵 → (𝐶 ∈ [𝑥]𝑅𝐶𝐵))
3 eqeq1 2735 . . . 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 elecALTV 36939 . . . . . 6 ((𝑥 ∈ V ∧ 𝐶 ∈ [𝑥]𝑅) → (𝐶 ∈ [𝑥]𝑅𝑥𝑅𝐶))
65el2v1 36890 . . . . 5 (𝐶 ∈ [𝑥]𝑅 → (𝐶 ∈ [𝑥]𝑅𝑥𝑅𝐶))
76ibi 266 . . . 4 (𝐶 ∈ [𝑥]𝑅𝑥𝑅𝐶)
8 simpll 765 . . . . . 6 ((( EqvRel 𝑅𝑥𝐴) ∧ 𝑥𝑅𝐶) → EqvRel 𝑅)
9 simpr 485 . . . . . 6 ((( EqvRel 𝑅𝑥𝐴) ∧ 𝑥𝑅𝐶) → 𝑥𝑅𝐶)
108, 9eqvrelthi 37288 . . . . 5 ((( EqvRel 𝑅𝑥𝐴) ∧ 𝑥𝑅𝐶) → [𝑥]𝑅 = [𝐶]𝑅)
1110ex 413 . . . 4 (( EqvRel 𝑅𝑥𝐴) → (𝑥𝑅𝐶 → [𝑥]𝑅 = [𝐶]𝑅))
127, 11syl5 34 . . 3 (( EqvRel 𝑅𝑥𝐴) → (𝐶 ∈ [𝑥]𝑅 → [𝑥]𝑅 = [𝐶]𝑅))
131, 4, 12ectocld 8761 . 2 (( EqvRel 𝑅𝐵 ∈ (𝐴 / 𝑅)) → (𝐶𝐵𝐵 = [𝐶]𝑅))
14133impia 1117 1 (( EqvRel 𝑅𝐵 ∈ (𝐴 / 𝑅) ∧ 𝐶𝐵) → 𝐵 = [𝐶]𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  Vcvv 3473   class class class wbr 5141  [cec 8684   / cqs 8685   EqvRel weqvrel 36865
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ec 8688  df-qs 8692  df-refrel 37187  df-symrel 37219  df-trrel 37249  df-eqvrel 37260
This theorem is referenced by:  erimeq2  37353
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