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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 3448 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
2 | 1 | ecid 8680 | . . . . . 6 ⊢ [𝑥]◡ E = 𝑥 |
3 | 2 | eqeq2i 2751 | . . . . 5 ⊢ (𝑦 = [𝑥]◡ E ↔ 𝑦 = 𝑥) |
4 | equcom 2022 | . . . . 5 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦) | |
5 | 3, 4 | bitri 275 | . . . 4 ⊢ (𝑦 = [𝑥]◡ E ↔ 𝑥 = 𝑦) |
6 | 5 | rexbii 3096 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = [𝑥]◡ E ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝑦) |
7 | vex 3448 | . . . 4 ⊢ 𝑦 ∈ V | |
8 | 7 | elqs 8667 | . . 3 ⊢ (𝑦 ∈ (𝐴 / ◡ E ) ↔ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]◡ E ) |
9 | risset 3220 | . . 3 ⊢ (𝑦 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝑦) | |
10 | 6, 8, 9 | 3bitr4i 303 | . 2 ⊢ (𝑦 ∈ (𝐴 / ◡ E ) ↔ 𝑦 ∈ 𝐴) |
11 | 10 | eqriv 2735 | 1 ⊢ (𝐴 / ◡ E ) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 ∃wrex 3072 E cep 5535 ◡ccnv 5631 [cec 8605 / cqs 8606 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pr 5383 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2816 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-br 5105 df-opab 5167 df-eprel 5536 df-xp 5638 df-cnv 5640 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-ec 8609 df-qs 8613 |
This theorem is referenced by: dfcnqs 11037 cnvepima 36730 n0elim 37044 |
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