Theorem dfpart2 39246

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 39243	. 2 ⊢ (𝑅 Part 𝐴 ↔ ( Disj 𝑅 ∧ 𝑅 DomainQs 𝐴))
2		df-dmqs 39097	. . 3 ⊢ (𝑅 DomainQs 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴)
3	2	anbi2i 629	. 2 ⊢ (( Disj 𝑅 ∧ 𝑅 DomainQs 𝐴) ↔ ( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴))
4	1, 3	bitri 276	1 ⊢ (𝑅 Part 𝐴 ↔ ( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴))

Syntax hints:   ↔ wb 207   ∧ wa 396   = wceq 1547  dom cdm 5625   / cqs 8639   DomainQs wdmqs 38581   Disj wdisjALTV 38593   Part wpart 38598

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 208  df-an 397  df-dmqs 39097  df-part 39243

This theorem is referenced by:  parteq1  39251  parteq2  39252  partim  39285  pet0  39292  petid  39294  petidres  39296  petinidres  39298  petxrnidres  39300  petincnvepres  39337  pet  39339