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Metamath Proof Explorer

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 ) )