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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 3459 | . . . 4 ⊢ (𝐴 ∈ [𝐵]𝑅 → 𝐴 ∈ V) | |
2 | ecexr 8277 | . . . 4 ⊢ (𝐴 ∈ [𝐵]𝑅 → 𝐵 ∈ V) | |
3 | 1, 2 | jca 515 | . . 3 ⊢ (𝐴 ∈ [𝐵]𝑅 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
4 | 3 | adantl 485 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ [𝐵]𝑅) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
5 | brrelex12 5568 | . . 3 ⊢ ((Rel 𝑅 ∧ 𝐵𝑅𝐴) → (𝐵 ∈ V ∧ 𝐴 ∈ V)) | |
6 | 5 | ancomd 465 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐵𝑅𝐴) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
7 | elecg 8315 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴)) | |
8 | 4, 6, 7 | pm5.21nd 801 | 1 ⊢ (Rel 𝑅 → (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2111 Vcvv 3441 class class class wbr 5030 Rel wrel 5524 [cec 8270 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-ec 8274 |
This theorem is referenced by: eqgid 18324 tgptsmscls 22755 eqg0el 30977 pstmfval 31249 ismntop 31377 topfneec 33816 releleccnv 35678 elecres 35694 eleccnvep 35698 inecmo 35769 elecxrn 35798 elec1cnvxrn2 35805 eleccossin 35883 |
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