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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 3476 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
2 | 1 | ecid 8780 | . . . . . 6 ⊢ [𝑥]◡ E = 𝑥 |
3 | 2 | eqeq2i 2743 | . . . . 5 ⊢ (𝑦 = [𝑥]◡ E ↔ 𝑦 = 𝑥) |
4 | equcom 2019 | . . . . 5 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦) | |
5 | 3, 4 | bitri 274 | . . . 4 ⊢ (𝑦 = [𝑥]◡ E ↔ 𝑥 = 𝑦) |
6 | 5 | rexbii 3092 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = [𝑥]◡ E ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝑦) |
7 | vex 3476 | . . . 4 ⊢ 𝑦 ∈ V | |
8 | 7 | elqs 8767 | . . 3 ⊢ (𝑦 ∈ (𝐴 / ◡ E ) ↔ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]◡ E ) |
9 | risset 3228 | . . 3 ⊢ (𝑦 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝑦) | |
10 | 6, 8, 9 | 3bitr4i 302 | . 2 ⊢ (𝑦 ∈ (𝐴 / ◡ E ) ↔ 𝑦 ∈ 𝐴) |
11 | 10 | eqriv 2727 | 1 ⊢ (𝐴 / ◡ E ) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 ∈ wcel 2104 ∃wrex 3068 E cep 5580 ◡ccnv 5676 [cec 8705 / cqs 8706 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5150 df-opab 5212 df-eprel 5581 df-xp 5683 df-cnv 5685 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-ec 8709 df-qs 8713 |
This theorem is referenced by: dfcnqs 11141 cnvepima 37511 n0elim 37825 |
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