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Mirrors > Home > MPE Home > Th. List > qsid | Structured version Visualization version GIF version |
Description: A set is equal to its quotient set modulo the converse membership relation. (Note: the converse membership relation is not an equivalence relation.) (Contributed by NM, 13-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.) |
Ref | Expression |
---|---|
qsid | ⊢ (𝐴 / ◡ E ) = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3412 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
2 | 1 | ecid 8464 | . . . . . 6 ⊢ [𝑥]◡ E = 𝑥 |
3 | 2 | eqeq2i 2750 | . . . . 5 ⊢ (𝑦 = [𝑥]◡ E ↔ 𝑦 = 𝑥) |
4 | equcom 2026 | . . . . 5 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦) | |
5 | 3, 4 | bitri 278 | . . . 4 ⊢ (𝑦 = [𝑥]◡ E ↔ 𝑥 = 𝑦) |
6 | 5 | rexbii 3170 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = [𝑥]◡ E ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝑦) |
7 | vex 3412 | . . . 4 ⊢ 𝑦 ∈ V | |
8 | 7 | elqs 8451 | . . 3 ⊢ (𝑦 ∈ (𝐴 / ◡ E ) ↔ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]◡ E ) |
9 | risset 3186 | . . 3 ⊢ (𝑦 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝑦) | |
10 | 6, 8, 9 | 3bitr4i 306 | . 2 ⊢ (𝑦 ∈ (𝐴 / ◡ E ) ↔ 𝑦 ∈ 𝐴) |
11 | 10 | eqriv 2734 | 1 ⊢ (𝐴 / ◡ E ) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2110 ∃wrex 3062 E cep 5459 ◡ccnv 5550 [cec 8389 / cqs 8390 |
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-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-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-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-br 5054 df-opab 5116 df-eprel 5460 df-xp 5557 df-cnv 5559 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-ec 8393 df-qs 8397 |
This theorem is referenced by: dfcnqs 10756 cnvepima 36209 n0el3 36500 |
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