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Mirrors > Home > MPE Home > Th. List > relelec | Structured version Visualization version GIF version |
Description: Membership in an equivalence class when 𝑅 is a relation. (Contributed by Mario Carneiro, 11-Sep-2015.) |
Ref | Expression |
---|---|
relelec | ⊢ (Rel 𝑅 → (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3487 | . . . 4 ⊢ (𝐴 ∈ [𝐵]𝑅 → 𝐴 ∈ V) | |
2 | ecexr 8707 | . . . 4 ⊢ (𝐴 ∈ [𝐵]𝑅 → 𝐵 ∈ V) | |
3 | 1, 2 | jca 511 | . . 3 ⊢ (𝐴 ∈ [𝐵]𝑅 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
4 | 3 | adantl 481 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ [𝐵]𝑅) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
5 | brrelex12 5721 | . . 3 ⊢ ((Rel 𝑅 ∧ 𝐵𝑅𝐴) → (𝐵 ∈ V ∧ 𝐴 ∈ V)) | |
6 | 5 | ancomd 461 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐵𝑅𝐴) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
7 | elecg 8745 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴)) | |
8 | 4, 6, 7 | pm5.21nd 799 | 1 ⊢ (Rel 𝑅 → (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∈ wcel 2098 Vcvv 3468 class class class wbr 5141 Rel wrel 5674 [cec 8700 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5142 df-opab 5204 df-xp 5675 df-rel 5676 df-cnv 5677 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ec 8704 |
This theorem is referenced by: eqgid 19105 tgptsmscls 24005 eqg0el 32980 pstmfval 33406 ismntop 33536 topfneec 35748 releleccnv 37636 elecres 37656 eleccnvep 37660 inecmo 37735 elecxrn 37767 elec1cnvxrn2 37778 eleccossin 37864 |
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