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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 2737 | . . 3 ⊢ (𝐴 / 𝑅) = (𝐴 / 𝑅) | |
2 | eleq2 2826 | . . . 4 ⊢ ([𝑥]𝑅 = 𝐵 → (𝐶 ∈ [𝑥]𝑅 ↔ 𝐶 ∈ 𝐵)) | |
3 | eqeq1 2741 | . . . 4 ⊢ ([𝑥]𝑅 = 𝐵 → ([𝑥]𝑅 = [𝐶]𝑅 ↔ 𝐵 = [𝐶]𝑅)) | |
4 | 2, 3 | imbi12d 348 | . . 3 ⊢ ([𝑥]𝑅 = 𝐵 → ((𝐶 ∈ [𝑥]𝑅 → [𝑥]𝑅 = [𝐶]𝑅) ↔ (𝐶 ∈ 𝐵 → 𝐵 = [𝐶]𝑅))) |
5 | elecALTV 36142 | . . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝐶 ∈ [𝑥]𝑅) → (𝐶 ∈ [𝑥]𝑅 ↔ 𝑥𝑅𝐶)) | |
6 | 5 | el2v1 36107 | . . . . 5 ⊢ (𝐶 ∈ [𝑥]𝑅 → (𝐶 ∈ [𝑥]𝑅 ↔ 𝑥𝑅𝐶)) |
7 | 6 | ibi 270 | . . . 4 ⊢ (𝐶 ∈ [𝑥]𝑅 → 𝑥𝑅𝐶) |
8 | simpll 767 | . . . . . 6 ⊢ ((( EqvRel 𝑅 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥𝑅𝐶) → EqvRel 𝑅) | |
9 | simpr 488 | . . . . . 6 ⊢ ((( EqvRel 𝑅 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥𝑅𝐶) → 𝑥𝑅𝐶) | |
10 | 8, 9 | eqvrelthi 36463 | . . . . 5 ⊢ ((( EqvRel 𝑅 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥𝑅𝐶) → [𝑥]𝑅 = [𝐶]𝑅) |
11 | 10 | ex 416 | . . . 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 8466 | . 2 ⊢ (( EqvRel 𝑅 ∧ 𝐵 ∈ (𝐴 / 𝑅)) → (𝐶 ∈ 𝐵 → 𝐵 = [𝐶]𝑅)) |
14 | 13 | 3impia 1119 | 1 ⊢ (( EqvRel 𝑅 ∧ 𝐵 ∈ (𝐴 / 𝑅) ∧ 𝐶 ∈ 𝐵) → 𝐵 = [𝐶]𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 Vcvv 3408 class class class wbr 5053 [cec 8389 / cqs 8390 EqvRel weqvrel 36087 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-br 5054 df-opab 5116 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-ec 8393 df-qs 8397 df-refrel 36367 df-symrel 36395 df-trrel 36425 df-eqvrel 36435 |
This theorem is referenced by: erim2 36526 |
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