Theorem dfpart2 38754

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 38751	. 2 ⊢ (𝑅 Part 𝐴 ↔ ( Disj 𝑅 ∧ 𝑅 DomainQs 𝐴))
2		df-dmqs 38624	. . 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 1540  dom cdm 5646   / cqs 8681   DomainQs wdmqs 38190   Disj wdisjALTV 38200   Part wpart 38205

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 38624  df-part 38751

This theorem is referenced by:  parteq1  38759  parteq2  38760  partim  38793  pet0  38800  petid  38802  petidres  38804  petinidres  38806  petxrnidres  38808  petincnvepres  38834  pet  38836