Theorem ecun

Description: The union coset of A . (Contributed by Peter Mazsa, 28-Jan-2026)

	Ref	Expression
Assertion	ecun	⊢ ( 𝐴 ∈ 𝑉 → [ 𝐴 ] ( 𝑅 ∪ 𝑆 ) = ( [ 𝐴 ] 𝑅 ∪ [ 𝐴 ] 𝑆 ) )

Proof

Step	Hyp	Ref	Expression
1		unab	⊢ ( { 𝑥 ∣ 𝐴 𝑅 𝑥 } ∪ { 𝑥 ∣ 𝐴 𝑆 𝑥 } ) = { 𝑥 ∣ ( 𝐴 𝑅 𝑥 ∨ 𝐴 𝑆 𝑥 ) }
2	1	a1i	⊢ ( 𝐴 ∈ 𝑉 → ( { 𝑥 ∣ 𝐴 𝑅 𝑥 } ∪ { 𝑥 ∣ 𝐴 𝑆 𝑥 } ) = { 𝑥 ∣ ( 𝐴 𝑅 𝑥 ∨ 𝐴 𝑆 𝑥 ) } )
3		dfec2	⊢ ( 𝐴 ∈ 𝑉 → [ 𝐴 ] 𝑅 = { 𝑥 ∣ 𝐴 𝑅 𝑥 } )
4		dfec2	⊢ ( 𝐴 ∈ 𝑉 → [ 𝐴 ] 𝑆 = { 𝑥 ∣ 𝐴 𝑆 𝑥 } )
5	3 4	uneq12d	⊢ ( 𝐴 ∈ 𝑉 → ( [ 𝐴 ] 𝑅 ∪ [ 𝐴 ] 𝑆 ) = ( { 𝑥 ∣ 𝐴 𝑅 𝑥 } ∪ { 𝑥 ∣ 𝐴 𝑆 𝑥 } ) )
6		elecALTV	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑥 ∈ V ) → ( 𝑥 ∈ [ 𝐴 ] ( 𝑅 ∪ 𝑆 ) ↔ 𝐴 ( 𝑅 ∪ 𝑆 ) 𝑥 ) )
7	6	elvd	⊢ ( 𝐴 ∈ 𝑉 → ( 𝑥 ∈ [ 𝐴 ] ( 𝑅 ∪ 𝑆 ) ↔ 𝐴 ( 𝑅 ∪ 𝑆 ) 𝑥 ) )
8		brun	⊢ ( 𝐴 ( 𝑅 ∪ 𝑆 ) 𝑥 ↔ ( 𝐴 𝑅 𝑥 ∨ 𝐴 𝑆 𝑥 ) )
9	7 8	bitrdi	⊢ ( 𝐴 ∈ 𝑉 → ( 𝑥 ∈ [ 𝐴 ] ( 𝑅 ∪ 𝑆 ) ↔ ( 𝐴 𝑅 𝑥 ∨ 𝐴 𝑆 𝑥 ) ) )
10	9	eqabdv	⊢ ( 𝐴 ∈ 𝑉 → [ 𝐴 ] ( 𝑅 ∪ 𝑆 ) = { 𝑥 ∣ ( 𝐴 𝑅 𝑥 ∨ 𝐴 𝑆 𝑥 ) } )
11	2 5 10	3eqtr4rd	⊢ ( 𝐴 ∈ 𝑉 → [ 𝐴 ] ( 𝑅 ∪ 𝑆 ) = ( [ 𝐴 ] 𝑅 ∪ [ 𝐴 ] 𝑆 ) )