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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 37337 | . . . 4 ⊢ (𝑅 ∈ 𝑊 → (𝑅 ∈ Disjs ↔ Disj 𝑅)) | |
2 | 1 | adantl 482 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑅 ∈ Disjs ↔ Disj 𝑅)) |
3 | brdmqssqs 37257 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑅 DomainQss 𝐴 ↔ 𝑅 DomainQs 𝐴)) | |
4 | 2, 3 | anbi12d 631 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ((𝑅 ∈ Disjs ∧ 𝑅 DomainQss 𝐴) ↔ ( Disj 𝑅 ∧ 𝑅 DomainQs 𝐴))) |
5 | brparts 37381 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑅 Parts 𝐴 ↔ (𝑅 ∈ Disjs ∧ 𝑅 DomainQss 𝐴))) | |
6 | 5 | adantr 481 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑅 Parts 𝐴 ↔ (𝑅 ∈ Disjs ∧ 𝑅 DomainQss 𝐴))) |
7 | df-part 37376 | . . 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 310 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑅 Parts 𝐴 ↔ 𝑅 Part 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∈ wcel 2106 class class class wbr 5136 DomainQss cdmqss 36806 DomainQs wdmqs 36807 Disjs cdisjs 36816 Disj wdisjALTV 36817 Parts cparts 36821 Part wpart 36822 |
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-10 2137 ax-12 2171 ax-ext 2702 ax-sep 5287 ax-nul 5294 ax-pow 5351 ax-pr 5415 ax-un 7703 |
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-nf 1786 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3429 df-v 3471 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4314 df-if 4518 df-pw 4593 df-sn 4618 df-pr 4620 df-op 4624 df-uni 4897 df-br 5137 df-opab 5199 df-xp 5670 df-rel 5671 df-cnv 5672 df-co 5673 df-dm 5674 df-rn 5675 df-res 5676 df-ima 5677 df-ec 8683 df-qs 8687 df-coss 37021 df-rels 37095 df-ssr 37108 df-cnvrefs 37135 df-cnvrefrels 37136 df-cnvrefrel 37137 df-dmqss 37248 df-dmqs 37249 df-disjss 37313 df-disjs 37314 df-disjALTV 37315 df-parts 37375 df-part 37376 |
This theorem is referenced by: mpets2 37451 pets 37462 |
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