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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 36932 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → ≀ ◡𝑅 ∈ V) | |
2 | elcnvrefrelsrel 37048 | . . . 4 ⊢ ( ≀ ◡𝑅 ∈ V → ( ≀ ◡𝑅 ∈ CnvRefRels ↔ CnvRefRel ≀ ◡𝑅)) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝑅 ∈ 𝑉 → ( ≀ ◡𝑅 ∈ CnvRefRels ↔ CnvRefRel ≀ ◡𝑅)) |
4 | elrelsrel 36999 | . . 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 37234 | . 2 ⊢ (𝑅 ∈ Disjs ↔ ( ≀ ◡𝑅 ∈ CnvRefRels ∧ 𝑅 ∈ Rels )) | |
7 | df-disjALTV 37217 | . 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 3447 ◡ccnv 5636 Rel wrel 5642 ≀ ccoss 36684 Rels crels 36686 CnvRefRels ccnvrefrels 36692 CnvRefRel wcnvrefrel 36693 Disjs cdisjs 36717 Disj wdisjALTV 36718 |
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 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
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 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-coss 36923 df-rels 36997 df-ssr 37010 df-cnvrefs 37037 df-cnvrefrels 37038 df-cnvrefrel 37039 df-disjss 37215 df-disjs 37216 df-disjALTV 37217 |
This theorem is referenced by: eleldisjseldisj 37241 brpartspart 37285 |
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