Theorem dfpart2 39897

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 39894	. 2 ⊢ (𝑅 Part 𝐴 ↔ ( Disj 𝑅 ∧ 𝑅 DomainQs 𝐴))
2		df-dmqs 39748	. . 3 ⊢ (𝑅 DomainQs 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴)
3	2	anbi2i 635	. 2 ⊢ (( Disj 𝑅 ∧ 𝑅 DomainQs 𝐴) ↔ ( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴))
4	1, 3	bitri 278	1 ⊢ (𝑅 Part 𝐴 ↔ ( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1569  dom cdm 5650   / cqs 8719   DomainQs wdmqs 39232   Disj wdisjALTV 39244   Part wpart 39249

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-dmqs 39748  df-part 39894

This theorem is used by:  parteq1  39902  parteq2  39903  partim  39936  pet0  39943  petid  39945  petidres  39947  petinidres  39949  petxrnidres  39951  petincnvepres  39988  pet  39990