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Mirrors > Home > MPE Home > Th. List > Mathboxes > elrels2 | Structured version Visualization version GIF version |
Description: The element of the relations class (df-rels 37867) and the relation predicate (df-rel 5676) are the same when 𝑅 is a set. (Contributed by Peter Mazsa, 14-Jun-2018.) |
Ref | Expression |
---|---|
elrels2 | ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ 𝑅 ⊆ (V × V))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rels 37867 | . . 3 ⊢ Rels = 𝒫 (V × V) | |
2 | 1 | eleq2i 2819 | . 2 ⊢ (𝑅 ∈ Rels ↔ 𝑅 ∈ 𝒫 (V × V)) |
3 | elpwg 4600 | . 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ 𝒫 (V × V) ↔ 𝑅 ⊆ (V × V))) | |
4 | 2, 3 | bitrid 283 | 1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ 𝑅 ⊆ (V × V))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2098 Vcvv 3468 ⊆ wss 3943 𝒫 cpw 4597 × cxp 5667 Rels crels 37557 |
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-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-v 3470 df-in 3950 df-ss 3960 df-pw 4599 df-rels 37867 |
This theorem is referenced by: elrelsrel 37869 |
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