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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 35699 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → ≀ ◡𝑅 ∈ V) | |
2 | elcnvrefrelsrel 35806 | . . . 4 ⊢ ( ≀ ◡𝑅 ∈ V → ( ≀ ◡𝑅 ∈ CnvRefRels ↔ CnvRefRel ≀ ◡𝑅)) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝑅 ∈ 𝑉 → ( ≀ ◡𝑅 ∈ CnvRefRels ↔ CnvRefRel ≀ ◡𝑅)) |
4 | elrelsrel 35761 | . . 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 35989 | . 2 ⊢ (𝑅 ∈ Disjs ↔ ( ≀ ◡𝑅 ∈ CnvRefRels ∧ 𝑅 ∈ Rels )) | |
7 | df-disjALTV 35972 | . 2 ⊢ ( Disj 𝑅 ↔ ( CnvRefRel ≀ ◡𝑅 ∧ Rel 𝑅)) | |
8 | 5, 6, 7 | 3bitr4g 316 | 1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Disjs ↔ Disj 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2113 Vcvv 3491 ◡ccnv 5547 Rel wrel 5553 ≀ ccoss 35487 Rels crels 35489 CnvRefRels ccnvrefrels 35495 CnvRefRel wcnvrefrel 35496 Disjs cdisjs 35520 Disj wdisjALTV 35521 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-coss 35693 df-rels 35759 df-ssr 35772 df-cnvrefs 35797 df-cnvrefrels 35798 df-cnvrefrel 35799 df-disjss 35970 df-disjs 35971 df-disjALTV 35972 |
This theorem is referenced by: eleldisjseldisj 35996 |
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