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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 35727) 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 35728 | . 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ 𝑅 ⊆ (V × V))) | |
2 | df-rel 5564 | . 2 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V)) | |
3 | 1, 2 | syl6bbr 291 | 1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∈ wcel 2114 Vcvv 3496 ⊆ wss 3938 × cxp 5555 Rel wrel 5562 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-rel 5564 df-rels 35727 |
This theorem is referenced by: elrelsrelim 35730 elrels5 35731 elrels6 35732 cnvelrels 35737 cosselrels 35738 elrefrelsrel 35761 elcnvrefrelsrel 35774 elsymrelsrel 35795 eltrrelsrel 35819 eleqvrelsrel 35831 elfunsALTVfunALTV 35932 eldisjs5 35961 eldisjsdisj 35962 |
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