![]() |
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 37290 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → ≀ ◡𝑅 ∈ V) | |
2 | elcnvrefrelsrel 37406 | . . . 4 ⊢ ( ≀ ◡𝑅 ∈ V → ( ≀ ◡𝑅 ∈ CnvRefRels ↔ CnvRefRel ≀ ◡𝑅)) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝑅 ∈ 𝑉 → ( ≀ ◡𝑅 ∈ CnvRefRels ↔ CnvRefRel ≀ ◡𝑅)) |
4 | elrelsrel 37357 | . . 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 37592 | . 2 ⊢ (𝑅 ∈ Disjs ↔ ( ≀ ◡𝑅 ∈ CnvRefRels ∧ 𝑅 ∈ Rels )) | |
7 | df-disjALTV 37575 | . 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 205 ∧ wa 397 ∈ wcel 2107 Vcvv 3475 ◡ccnv 5676 Rel wrel 5682 ≀ ccoss 37043 Rels crels 37045 CnvRefRels ccnvrefrels 37051 CnvRefRel wcnvrefrel 37052 Disjs cdisjs 37076 Disj wdisjALTV 37077 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-coss 37281 df-rels 37355 df-ssr 37368 df-cnvrefs 37395 df-cnvrefrels 37396 df-cnvrefrel 37397 df-disjss 37573 df-disjs 37574 df-disjALTV 37575 |
This theorem is referenced by: eleldisjseldisj 37599 brpartspart 37643 |
Copyright terms: Public domain | W3C validator |