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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 35727) and the relation predicate (df-rel 5564) 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 35727 | . . 3 ⊢ Rels = 𝒫 (V × V) | |
2 | 1 | eleq2i 2906 | . 2 ⊢ (𝑅 ∈ Rels ↔ 𝑅 ∈ 𝒫 (V × V)) |
3 | elpwg 4544 | . 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ 𝒫 (V × V) ↔ 𝑅 ⊆ (V × V))) | |
4 | 2, 3 | syl5bb 285 | 1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ 𝑅 ⊆ (V × V))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∈ wcel 2114 Vcvv 3496 ⊆ wss 3938 𝒫 cpw 4541 × cxp 5555 Rels crels 35457 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-in 3945 df-ss 3954 df-pw 4543 df-rels 35727 |
This theorem is referenced by: elrelsrel 35729 |
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