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Mathbox for Peter Mazsa |
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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 38402 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → ≀ ◡𝑅 ∈ V) | |
2 | elcnvrefrelsrel 38518 | . . . 4 ⊢ ( ≀ ◡𝑅 ∈ V → ( ≀ ◡𝑅 ∈ CnvRefRels ↔ CnvRefRel ≀ ◡𝑅)) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝑅 ∈ 𝑉 → ( ≀ ◡𝑅 ∈ CnvRefRels ↔ CnvRefRel ≀ ◡𝑅)) |
4 | elrelsrel 38469 | . . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅)) | |
5 | 3, 4 | anbi12d 632 | . 2 ⊢ (𝑅 ∈ 𝑉 → (( ≀ ◡𝑅 ∈ CnvRefRels ∧ 𝑅 ∈ Rels ) ↔ ( CnvRefRel ≀ ◡𝑅 ∧ Rel 𝑅))) |
6 | eldisjs 38704 | . 2 ⊢ (𝑅 ∈ Disjs ↔ ( ≀ ◡𝑅 ∈ CnvRefRels ∧ 𝑅 ∈ Rels )) | |
7 | df-disjALTV 38687 | . 2 ⊢ ( Disj 𝑅 ↔ ( CnvRefRel ≀ ◡𝑅 ∧ Rel 𝑅)) | |
8 | 5, 6, 7 | 3bitr4g 314 | 1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Disjs ↔ Disj 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2106 Vcvv 3478 ◡ccnv 5688 Rel wrel 5694 ≀ ccoss 38162 Rels crels 38164 CnvRefRels ccnvrefrels 38170 CnvRefRel wcnvrefrel 38171 Disjs cdisjs 38195 Disj wdisjALTV 38196 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-coss 38393 df-rels 38467 df-ssr 38480 df-cnvrefs 38507 df-cnvrefrels 38508 df-cnvrefrel 38509 df-disjss 38685 df-disjs 38686 df-disjALTV 38687 |
This theorem is referenced by: eleldisjseldisj 38711 brpartspart 38755 |
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