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Theorem eqvrelqsel 38144
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 2725 . . 3 (𝐴 / 𝑅) = (𝐴 / 𝑅)
2 eleq2 2814 . . . 4 ([𝑥]𝑅 = 𝐵 → (𝐶 ∈ [𝑥]𝑅𝐶𝐵))
3 eqeq1 2729 . . . 4 ([𝑥]𝑅 = 𝐵 → ([𝑥]𝑅 = [𝐶]𝑅𝐵 = [𝐶]𝑅))
42, 3imbi12d 343 . . 3 ([𝑥]𝑅 = 𝐵 → ((𝐶 ∈ [𝑥]𝑅 → [𝑥]𝑅 = [𝐶]𝑅) ↔ (𝐶𝐵𝐵 = [𝐶]𝑅)))
5 elecALTV 37794 . . . . . 6 ((𝑥 ∈ V ∧ 𝐶 ∈ [𝑥]𝑅) → (𝐶 ∈ [𝑥]𝑅𝑥𝑅𝐶))
65el2v1 37747 . . . . 5 (𝐶 ∈ [𝑥]𝑅 → (𝐶 ∈ [𝑥]𝑅𝑥𝑅𝐶))
76ibi 266 . . . 4 (𝐶 ∈ [𝑥]𝑅𝑥𝑅𝐶)
8 simpll 765 . . . . . 6 ((( EqvRel 𝑅𝑥𝐴) ∧ 𝑥𝑅𝐶) → EqvRel 𝑅)
9 simpr 483 . . . . . 6 ((( EqvRel 𝑅𝑥𝐴) ∧ 𝑥𝑅𝐶) → 𝑥𝑅𝐶)
108, 9eqvrelthi 38141 . . . . 5 ((( EqvRel 𝑅𝑥𝐴) ∧ 𝑥𝑅𝐶) → [𝑥]𝑅 = [𝐶]𝑅)
1110ex 411 . . . 4 (( EqvRel 𝑅𝑥𝐴) → (𝑥𝑅𝐶 → [𝑥]𝑅 = [𝐶]𝑅))
127, 11syl5 34 . . 3 (( EqvRel 𝑅𝑥𝐴) → (𝐶 ∈ [𝑥]𝑅 → [𝑥]𝑅 = [𝐶]𝑅))
131, 4, 12ectocld 8801 . 2 (( EqvRel 𝑅𝐵 ∈ (𝐴 / 𝑅)) → (𝐶𝐵𝐵 = [𝐶]𝑅))
14133impia 1114 1 (( EqvRel 𝑅𝐵 ∈ (𝐴 / 𝑅) ∧ 𝐶𝐵) → 𝐵 = [𝐶]𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  Vcvv 3463   class class class wbr 5143  [cec 8721   / cqs 8722   EqvRel weqvrel 37722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ec 8725  df-qs 8729  df-refrel 38040  df-symrel 38072  df-trrel 38102  df-eqvrel 38113
This theorem is referenced by:  erimeq2  38206
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