Theorem reldisjun

Description: Split a relation into two disjoint parts based on its domain. (Contributed by Thierry Arnoux, 9-Oct-2023)

	Ref	Expression
Assertion	reldisjun	⊢ ( ( Rel 𝑅 ∧ dom 𝑅 = ( 𝐴 ∪ 𝐵 ) ∧ ( 𝐴 ∩ 𝐵 ) = ∅ ) → 𝑅 = ( ( 𝑅 ↾ 𝐴 ) ∪ ( 𝑅 ↾ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		reseq2	⊢ ( dom 𝑅 = ( 𝐴 ∪ 𝐵 ) → ( 𝑅 ↾ dom 𝑅 ) = ( 𝑅 ↾ ( 𝐴 ∪ 𝐵 ) ) )
2	1	3ad2ant2	⊢ ( ( Rel 𝑅 ∧ dom 𝑅 = ( 𝐴 ∪ 𝐵 ) ∧ ( 𝐴 ∩ 𝐵 ) = ∅ ) → ( 𝑅 ↾ dom 𝑅 ) = ( 𝑅 ↾ ( 𝐴 ∪ 𝐵 ) ) )
3		resdm	⊢ ( Rel 𝑅 → ( 𝑅 ↾ dom 𝑅 ) = 𝑅 )
4	3	3ad2ant1	⊢ ( ( Rel 𝑅 ∧ dom 𝑅 = ( 𝐴 ∪ 𝐵 ) ∧ ( 𝐴 ∩ 𝐵 ) = ∅ ) → ( 𝑅 ↾ dom 𝑅 ) = 𝑅 )
5		resundi	⊢ ( 𝑅 ↾ ( 𝐴 ∪ 𝐵 ) ) = ( ( 𝑅 ↾ 𝐴 ) ∪ ( 𝑅 ↾ 𝐵 ) )
6	5	a1i	⊢ ( ( Rel 𝑅 ∧ dom 𝑅 = ( 𝐴 ∪ 𝐵 ) ∧ ( 𝐴 ∩ 𝐵 ) = ∅ ) → ( 𝑅 ↾ ( 𝐴 ∪ 𝐵 ) ) = ( ( 𝑅 ↾ 𝐴 ) ∪ ( 𝑅 ↾ 𝐵 ) ) )
7	2 4 6	3eqtr3d	⊢ ( ( Rel 𝑅 ∧ dom 𝑅 = ( 𝐴 ∪ 𝐵 ) ∧ ( 𝐴 ∩ 𝐵 ) = ∅ ) → 𝑅 = ( ( 𝑅 ↾ 𝐴 ) ∪ ( 𝑅 ↾ 𝐵 ) ) )