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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 3484 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
| 2 | 1 | ecid 8822 | . . . . . 6 ⊢ [𝑥]◡ E = 𝑥 | 
| 3 | 2 | eqeq2i 2750 | . . . . 5 ⊢ (𝑦 = [𝑥]◡ E ↔ 𝑦 = 𝑥) | 
| 4 | equcom 2017 | . . . . 5 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦) | |
| 5 | 3, 4 | bitri 275 | . . . 4 ⊢ (𝑦 = [𝑥]◡ E ↔ 𝑥 = 𝑦) | 
| 6 | 5 | rexbii 3094 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = [𝑥]◡ E ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝑦) | 
| 7 | vex 3484 | . . . 4 ⊢ 𝑦 ∈ V | |
| 8 | 7 | elqs 8809 | . . 3 ⊢ (𝑦 ∈ (𝐴 / ◡ E ) ↔ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]◡ E ) | 
| 9 | risset 3233 | . . 3 ⊢ (𝑦 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝑦) | |
| 10 | 6, 8, 9 | 3bitr4i 303 | . 2 ⊢ (𝑦 ∈ (𝐴 / ◡ E ) ↔ 𝑦 ∈ 𝐴) | 
| 11 | 10 | eqriv 2734 | 1 ⊢ (𝐴 / ◡ E ) = 𝐴 | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ∈ wcel 2108 ∃wrex 3070 E cep 5583 ◡ccnv 5684 [cec 8743 / cqs 8744 | 
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-eprel 5584 df-xp 5691 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ec 8747 df-qs 8751 | 
| This theorem is referenced by: dfcnqs 11182 cnvepima 38338 n0elim 38651 | 
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