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Theorem dfpart2 37634
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 37631 . 2 (𝑅 Part 𝐴 ↔ ( Disj 𝑅𝑅 DomainQs 𝐴))
2 df-dmqs 37504 . . 3 (𝑅 DomainQs 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴)
32anbi2i 623 . 2 (( Disj 𝑅𝑅 DomainQs 𝐴) ↔ ( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴))
41, 3bitri 274 1 (𝑅 Part 𝐴 ↔ ( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1541  dom cdm 5676   / cqs 8701   DomainQs wdmqs 37062   Disj wdisjALTV 37072   Part wpart 37077
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 397  df-dmqs 37504  df-part 37631
This theorem is referenced by:  parteq1  37639  parteq2  37640  partim  37673  pet0  37680  petid  37682  petidres  37684  petinidres  37686  petxrnidres  37688  petincnvepres  37714  pet  37716
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