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| Mirrors > Home > MPE Home > Th. List > Mathboxes > elrelsrel | Structured version Visualization version GIF version | ||
| Description: The element of the relations class (df-rels 38486) and the relation predicate are the same when 𝑅 is a set. (Contributed by Peter Mazsa, 24-Nov-2018.) | 
| Ref | Expression | 
|---|---|
| elrelsrel | ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elrels2 38487 | . 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ 𝑅 ⊆ (V × V))) | |
| 2 | df-rel 5692 | . 2 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V)) | |
| 3 | 1, 2 | bitr4di 289 | 1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 × cxp 5683 Rel wrel 5690 Rels crels 38184 | 
| 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 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ss 3968 df-pw 4602 df-rel 5692 df-rels 38486 | 
| This theorem is referenced by: elrelsrelim 38489 elrels5 38490 elrels6 38491 cnvelrels 38496 cosselrels 38497 elrefrelsrel 38521 elcnvrefrelsrel 38537 elsymrelsrel 38558 eltrrelsrel 38582 eleqvrelsrel 38595 elfunsALTVfunALTV 38698 eldisjs5 38727 eldisjsdisj 38728 | 
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