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Theorem dfpart2 38151
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 38148 . 2 (𝑅 Part 𝐴 ↔ ( Disj 𝑅𝑅 DomainQs 𝐴))
2 df-dmqs 38021 . . 3 (𝑅 DomainQs 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴)
32anbi2i 622 . 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 395   = wceq 1533  dom cdm 5669   / cqs 8701   DomainQs wdmqs 37579   Disj wdisjALTV 37589   Part wpart 37594
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 396  df-dmqs 38021  df-part 38148
This theorem is referenced by:  parteq1  38156  parteq2  38157  partim  38190  pet0  38197  petid  38199  petidres  38201  petinidres  38203  petxrnidres  38205  petincnvepres  38231  pet  38233
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