Theorem dfpart2 38751

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

Step	Hyp	Ref	Expression
1		df-part 38748	. 2 ⊢ (𝑅 Part 𝐴 ↔ ( Disj 𝑅 ∧ 𝑅 DomainQs 𝐴))
2		df-dmqs 38621	. . 3 ⊢ (𝑅 DomainQs 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴)
3	2	anbi2i 623	. 2 ⊢ (( Disj 𝑅 ∧ 𝑅 DomainQs 𝐴) ↔ ( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴))
4	1, 3	bitri 275	1 ⊢ (𝑅 Part 𝐴 ↔ ( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1537  dom cdm 5689   / cqs 8743   DomainQs wdmqs 38186   Disj wdisjALTV 38196   Part wpart 38201

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 207  df-an 396  df-dmqs 38621  df-part 38748

This theorem is referenced by:  parteq1  38756  parteq2  38757  partim  38790  pet0  38797  petid  38799  petidres  38801  petinidres  38803  petxrnidres  38805  petincnvepres  38831  pet  38833