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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 3461 | . . . 4 ⊢ (𝐴 ∈ [𝐵]𝑅 → 𝐴 ∈ V) | |
2 | ecexr 8611 | . . . 4 ⊢ (𝐴 ∈ [𝐵]𝑅 → 𝐵 ∈ V) | |
3 | 1, 2 | jca 512 | . . 3 ⊢ (𝐴 ∈ [𝐵]𝑅 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
4 | 3 | adantl 482 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ [𝐵]𝑅) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
5 | brrelex12 5682 | . . 3 ⊢ ((Rel 𝑅 ∧ 𝐵𝑅𝐴) → (𝐵 ∈ V ∧ 𝐴 ∈ V)) | |
6 | 5 | ancomd 462 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐵𝑅𝐴) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
7 | elecg 8649 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴)) | |
8 | 4, 6, 7 | pm5.21nd 800 | 1 ⊢ (Rel 𝑅 → (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∈ wcel 2106 Vcvv 3443 class class class wbr 5103 Rel wrel 5636 [cec 8604 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-br 5104 df-opab 5166 df-xp 5637 df-rel 5638 df-cnv 5639 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-ec 8608 |
This theorem is referenced by: eqgid 18940 tgptsmscls 23452 eqg0el 31972 pstmfval 32280 ismntop 32410 topfneec 34758 releleccnv 36648 elecres 36668 eleccnvep 36672 inecmo 36747 elecxrn 36779 elec1cnvxrn2 36790 eleccossin 36876 |
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