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Mirrors > Home > MPE Home > Th. List > Mathboxes > brparts2 | Structured version Visualization version GIF version |
Description: Binary partitions relation. (Contributed by Peter Mazsa, 30-Dec-2021.) |
Ref | Expression |
---|---|
brparts2 | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑅 Parts 𝐴 ↔ (𝑅 ∈ Disjs ∧ (dom 𝑅 / 𝑅) = 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brparts 37636 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑅 Parts 𝐴 ↔ (𝑅 ∈ Disjs ∧ 𝑅 DomainQss 𝐴))) | |
2 | 1 | adantr 481 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑅 Parts 𝐴 ↔ (𝑅 ∈ Disjs ∧ 𝑅 DomainQss 𝐴))) |
3 | brdmqss 37511 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑅 DomainQss 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴)) | |
4 | 3 | anbi2d 629 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ((𝑅 ∈ Disjs ∧ 𝑅 DomainQss 𝐴) ↔ (𝑅 ∈ Disjs ∧ (dom 𝑅 / 𝑅) = 𝐴))) |
5 | 2, 4 | bitrd 278 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑅 Parts 𝐴 ↔ (𝑅 ∈ Disjs ∧ (dom 𝑅 / 𝑅) = 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 class class class wbr 5148 dom cdm 5676 / cqs 8701 DomainQss cdmqss 37061 Disjs cdisjs 37071 Parts cparts 37076 |
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-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
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-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-xp 5682 df-cnv 5684 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-ec 8704 df-qs 8708 df-dmqss 37503 df-parts 37630 |
This theorem is referenced by: (None) |
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