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| Mirrors > Home > MPE Home > Th. List > Mathboxes > brpartspart | Structured version Visualization version GIF version | ||
| Description: Binary partition and the partition predicate are the same if 𝐴 and 𝑅 are sets. (Contributed by Peter Mazsa, 5-Sep-2021.) | 
| Ref | Expression | 
|---|---|
| brpartspart | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑅 Parts 𝐴 ↔ 𝑅 Part 𝐴)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eldisjsdisj 38728 | . . . 4 ⊢ (𝑅 ∈ 𝑊 → (𝑅 ∈ Disjs ↔ Disj 𝑅)) | |
| 2 | 1 | adantl 481 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑅 ∈ Disjs ↔ Disj 𝑅)) | 
| 3 | brdmqssqs 38648 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑅 DomainQss 𝐴 ↔ 𝑅 DomainQs 𝐴)) | |
| 4 | 2, 3 | anbi12d 632 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ((𝑅 ∈ Disjs ∧ 𝑅 DomainQss 𝐴) ↔ ( Disj 𝑅 ∧ 𝑅 DomainQs 𝐴))) | 
| 5 | brparts 38772 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑅 Parts 𝐴 ↔ (𝑅 ∈ Disjs ∧ 𝑅 DomainQss 𝐴))) | |
| 6 | 5 | adantr 480 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑅 Parts 𝐴 ↔ (𝑅 ∈ Disjs ∧ 𝑅 DomainQss 𝐴))) | 
| 7 | df-part 38767 | . . 3 ⊢ (𝑅 Part 𝐴 ↔ ( Disj 𝑅 ∧ 𝑅 DomainQs 𝐴)) | |
| 8 | 7 | a1i 11 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑅 Part 𝐴 ↔ ( Disj 𝑅 ∧ 𝑅 DomainQs 𝐴))) | 
| 9 | 4, 6, 8 | 3bitr4d 311 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑅 Parts 𝐴 ↔ 𝑅 Part 𝐴)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2108 class class class wbr 5143 DomainQss cdmqss 38205 DomainQs wdmqs 38206 Disjs cdisjs 38215 Disj wdisjALTV 38216 Parts cparts 38220 Part wpart 38221 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ec 8747 df-qs 8751 df-coss 38412 df-rels 38486 df-ssr 38499 df-cnvrefs 38526 df-cnvrefrels 38527 df-cnvrefrel 38528 df-dmqss 38639 df-dmqs 38640 df-disjss 38704 df-disjs 38705 df-disjALTV 38706 df-parts 38766 df-part 38767 | 
| This theorem is referenced by: mpets2 38842 pets 38853 | 
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