Metamath Proof Explorer


Theorem rabiun

Description: Abstraction restricted to an indexed union. (Contributed by Brendan Leahy, 26-Oct-2017)

Ref Expression
Assertion rabiun { 𝑥 𝑦𝐴 𝐵𝜑 } = 𝑦𝐴 { 𝑥𝐵𝜑 }

Proof

Step Hyp Ref Expression
1 eliun ( 𝑥 𝑦𝐴 𝐵 ↔ ∃ 𝑦𝐴 𝑥𝐵 )
2 1 anbi1i ( ( 𝑥 𝑦𝐴 𝐵𝜑 ) ↔ ( ∃ 𝑦𝐴 𝑥𝐵𝜑 ) )
3 r19.41v ( ∃ 𝑦𝐴 ( 𝑥𝐵𝜑 ) ↔ ( ∃ 𝑦𝐴 𝑥𝐵𝜑 ) )
4 2 3 bitr4i ( ( 𝑥 𝑦𝐴 𝐵𝜑 ) ↔ ∃ 𝑦𝐴 ( 𝑥𝐵𝜑 ) )
5 4 abbii { 𝑥 ∣ ( 𝑥 𝑦𝐴 𝐵𝜑 ) } = { 𝑥 ∣ ∃ 𝑦𝐴 ( 𝑥𝐵𝜑 ) }
6 df-rab { 𝑥 𝑦𝐴 𝐵𝜑 } = { 𝑥 ∣ ( 𝑥 𝑦𝐴 𝐵𝜑 ) }
7 iunab 𝑦𝐴 { 𝑥 ∣ ( 𝑥𝐵𝜑 ) } = { 𝑥 ∣ ∃ 𝑦𝐴 ( 𝑥𝐵𝜑 ) }
8 5 6 7 3eqtr4i { 𝑥 𝑦𝐴 𝐵𝜑 } = 𝑦𝐴 { 𝑥 ∣ ( 𝑥𝐵𝜑 ) }
9 df-rab { 𝑥𝐵𝜑 } = { 𝑥 ∣ ( 𝑥𝐵𝜑 ) }
10 9 a1i ( 𝑦𝐴 → { 𝑥𝐵𝜑 } = { 𝑥 ∣ ( 𝑥𝐵𝜑 ) } )
11 10 iuneq2i 𝑦𝐴 { 𝑥𝐵𝜑 } = 𝑦𝐴 { 𝑥 ∣ ( 𝑥𝐵𝜑 ) }
12 8 11 eqtr4i { 𝑥 𝑦𝐴 𝐵𝜑 } = 𝑦𝐴 { 𝑥𝐵𝜑 }