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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 3497 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
2 | 1 | ecid 8362 | . . . . . 6 ⊢ [𝑥]◡ E = 𝑥 |
3 | 2 | eqeq2i 2834 | . . . . 5 ⊢ (𝑦 = [𝑥]◡ E ↔ 𝑦 = 𝑥) |
4 | equcom 2025 | . . . . 5 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦) | |
5 | 3, 4 | bitri 277 | . . . 4 ⊢ (𝑦 = [𝑥]◡ E ↔ 𝑥 = 𝑦) |
6 | 5 | rexbii 3247 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = [𝑥]◡ E ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝑦) |
7 | vex 3497 | . . . 4 ⊢ 𝑦 ∈ V | |
8 | 7 | elqs 8349 | . . 3 ⊢ (𝑦 ∈ (𝐴 / ◡ E ) ↔ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]◡ E ) |
9 | risset 3267 | . . 3 ⊢ (𝑦 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝑦) | |
10 | 6, 8, 9 | 3bitr4i 305 | . 2 ⊢ (𝑦 ∈ (𝐴 / ◡ E ) ↔ 𝑦 ∈ 𝐴) |
11 | 10 | eqriv 2818 | 1 ⊢ (𝐴 / ◡ E ) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 ∃wrex 3139 E cep 5464 ◡ccnv 5554 [cec 8287 / cqs 8288 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-eprel 5465 df-xp 5561 df-cnv 5563 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-ec 8291 df-qs 8295 |
This theorem is referenced by: dfcnqs 10564 cnvepima 35609 n0el3 35900 |
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