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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 35885) and the relation predicate (df-rel 5526) 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 35885 | . . 3 ⊢ Rels = 𝒫 (V × V) | |
2 | 1 | eleq2i 2881 | . 2 ⊢ (𝑅 ∈ Rels ↔ 𝑅 ∈ 𝒫 (V × V)) |
3 | elpwg 4500 | . 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ 𝒫 (V × V) ↔ 𝑅 ⊆ (V × V))) | |
4 | 2, 3 | syl5bb 286 | 1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ 𝑅 ⊆ (V × V))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∈ wcel 2111 Vcvv 3441 ⊆ wss 3881 𝒫 cpw 4497 × cxp 5517 Rels crels 35615 |
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-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-in 3888 df-ss 3898 df-pw 4499 df-rels 35885 |
This theorem is referenced by: elrelsrel 35887 |
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