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Mirrors > Home > MPE Home > Th. List > Mathboxes > eqvrelqsel | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
eqvrelqsel | ⊢ (( EqvRel 𝑅 ∧ 𝐵 ∈ (𝐴 / 𝑅) ∧ 𝐶 ∈ 𝐵) → 𝐵 = [𝐶]𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2726 | . . 3 ⊢ (𝐴 / 𝑅) = (𝐴 / 𝑅) | |
2 | eleq2 2816 | . . . 4 ⊢ ([𝑥]𝑅 = 𝐵 → (𝐶 ∈ [𝑥]𝑅 ↔ 𝐶 ∈ 𝐵)) | |
3 | eqeq1 2730 | . . . 4 ⊢ ([𝑥]𝑅 = 𝐵 → ([𝑥]𝑅 = [𝐶]𝑅 ↔ 𝐵 = [𝐶]𝑅)) | |
4 | 2, 3 | imbi12d 344 | . . 3 ⊢ ([𝑥]𝑅 = 𝐵 → ((𝐶 ∈ [𝑥]𝑅 → [𝑥]𝑅 = [𝐶]𝑅) ↔ (𝐶 ∈ 𝐵 → 𝐵 = [𝐶]𝑅))) |
5 | elecALTV 37647 | . . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝐶 ∈ [𝑥]𝑅) → (𝐶 ∈ [𝑥]𝑅 ↔ 𝑥𝑅𝐶)) | |
6 | 5 | el2v1 37598 | . . . . 5 ⊢ (𝐶 ∈ [𝑥]𝑅 → (𝐶 ∈ [𝑥]𝑅 ↔ 𝑥𝑅𝐶)) |
7 | 6 | ibi 267 | . . . 4 ⊢ (𝐶 ∈ [𝑥]𝑅 → 𝑥𝑅𝐶) |
8 | simpll 764 | . . . . . 6 ⊢ ((( EqvRel 𝑅 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥𝑅𝐶) → EqvRel 𝑅) | |
9 | simpr 484 | . . . . . 6 ⊢ ((( EqvRel 𝑅 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥𝑅𝐶) → 𝑥𝑅𝐶) | |
10 | 8, 9 | eqvrelthi 37996 | . . . . 5 ⊢ ((( EqvRel 𝑅 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥𝑅𝐶) → [𝑥]𝑅 = [𝐶]𝑅) |
11 | 10 | ex 412 | . . . 4 ⊢ (( EqvRel 𝑅 ∧ 𝑥 ∈ 𝐴) → (𝑥𝑅𝐶 → [𝑥]𝑅 = [𝐶]𝑅)) |
12 | 7, 11 | syl5 34 | . . 3 ⊢ (( EqvRel 𝑅 ∧ 𝑥 ∈ 𝐴) → (𝐶 ∈ [𝑥]𝑅 → [𝑥]𝑅 = [𝐶]𝑅)) |
13 | 1, 4, 12 | ectocld 8780 | . 2 ⊢ (( EqvRel 𝑅 ∧ 𝐵 ∈ (𝐴 / 𝑅)) → (𝐶 ∈ 𝐵 → 𝐵 = [𝐶]𝑅)) |
14 | 13 | 3impia 1114 | 1 ⊢ (( EqvRel 𝑅 ∧ 𝐵 ∈ (𝐴 / 𝑅) ∧ 𝐶 ∈ 𝐵) → 𝐵 = [𝐶]𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 Vcvv 3468 class class class wbr 5141 [cec 8703 / cqs 8704 EqvRel weqvrel 37573 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 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 8707 df-qs 8711 df-refrel 37895 df-symrel 37927 df-trrel 37957 df-eqvrel 37968 |
This theorem is referenced by: erimeq2 38061 |
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