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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 37120 | . . . 4 ⊢ (𝑅 ∈ 𝑊 → (𝑅 ∈ Disjs ↔ Disj 𝑅)) | |
2 | 1 | adantl 482 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑅 ∈ Disjs ↔ Disj 𝑅)) |
3 | brdmqssqs 37040 | . . 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 37164 | . . 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 37159 | . . 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 5103 DomainQss cdmqss 36588 DomainQs wdmqs 36589 Disjs cdisjs 36598 Disj wdisjALTV 36599 Parts cparts 36603 Part wpart 36604 |
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 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 |
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 2715 df-cleq 2729 df-clel 2815 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-ec 8608 df-qs 8612 df-coss 36804 df-rels 36878 df-ssr 36891 df-cnvrefs 36918 df-cnvrefrels 36919 df-cnvrefrel 36920 df-dmqss 37031 df-dmqs 37032 df-disjss 37096 df-disjs 37097 df-disjALTV 37098 df-parts 37158 df-part 37159 |
This theorem is referenced by: mpets2 37234 pets 37245 |
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