![]() |
Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > eldisjsdisj | Structured version Visualization version GIF version |
Description: The element of the class of disjoint relations and the disjoint relation predicate are the same, that is (𝑅 ∈ Disjs ↔ Disj 𝑅) when 𝑅 is a set. (Contributed by Peter Mazsa, 25-Jul-2021.) |
Ref | Expression |
---|---|
eldisjsdisj | ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Disjs ↔ Disj 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cosscnvex 37093 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → ≀ ◡𝑅 ∈ V) | |
2 | elcnvrefrelsrel 37209 | . . . 4 ⊢ ( ≀ ◡𝑅 ∈ V → ( ≀ ◡𝑅 ∈ CnvRefRels ↔ CnvRefRel ≀ ◡𝑅)) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝑅 ∈ 𝑉 → ( ≀ ◡𝑅 ∈ CnvRefRels ↔ CnvRefRel ≀ ◡𝑅)) |
4 | elrelsrel 37160 | . . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅)) | |
5 | 3, 4 | anbi12d 631 | . 2 ⊢ (𝑅 ∈ 𝑉 → (( ≀ ◡𝑅 ∈ CnvRefRels ∧ 𝑅 ∈ Rels ) ↔ ( CnvRefRel ≀ ◡𝑅 ∧ Rel 𝑅))) |
6 | eldisjs 37395 | . 2 ⊢ (𝑅 ∈ Disjs ↔ ( ≀ ◡𝑅 ∈ CnvRefRels ∧ 𝑅 ∈ Rels )) | |
7 | df-disjALTV 37378 | . 2 ⊢ ( Disj 𝑅 ↔ ( CnvRefRel ≀ ◡𝑅 ∧ Rel 𝑅)) | |
8 | 5, 6, 7 | 3bitr4g 313 | 1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Disjs ↔ Disj 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∈ wcel 2106 Vcvv 3473 ◡ccnv 5668 Rel wrel 5674 ≀ ccoss 36846 Rels crels 36848 CnvRefRels ccnvrefrels 36854 CnvRefRel wcnvrefrel 36855 Disjs cdisjs 36879 Disj wdisjALTV 36880 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-opab 5204 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-coss 37084 df-rels 37158 df-ssr 37171 df-cnvrefs 37198 df-cnvrefrels 37199 df-cnvrefrel 37200 df-disjss 37376 df-disjs 37377 df-disjALTV 37378 |
This theorem is referenced by: eleldisjseldisj 37402 brpartspart 37446 |
Copyright terms: Public domain | W3C validator |