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Theorem dfpart2 37577
Description: Alternate definition of the partition predicate. (Contributed by Peter Mazsa, 5-Sep-2021.)
Assertion
Ref Expression
dfpart2 (𝑅 Part 𝐴 ↔ ( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴))

Proof of Theorem dfpart2
StepHypRef Expression
1 df-part 37574 . 2 (𝑅 Part 𝐴 ↔ ( Disj 𝑅𝑅 DomainQs 𝐴))
2 df-dmqs 37447 . . 3 (𝑅 DomainQs 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴)
32anbi2i 624 . 2 (( Disj 𝑅𝑅 DomainQs 𝐴) ↔ ( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴))
41, 3bitri 275 1 (𝑅 Part 𝐴 ↔ ( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  dom cdm 5675   / cqs 8698   DomainQs wdmqs 37005   Disj wdisjALTV 37015   Part wpart 37020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 206  df-an 398  df-dmqs 37447  df-part 37574
This theorem is referenced by:  parteq1  37582  parteq2  37583  partim  37616  pet0  37623  petid  37625  petidres  37627  petinidres  37629  petxrnidres  37631  petincnvepres  37657  pet  37659
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