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Metamath Proof Explorer

Theorem imaundi

Description: Distributive law for image over union. Theorem 35 of Suppes p. 65. (Contributed by NM, 30-Sep-2002)

Ref Expression
Assertion imaundi ( 𝐴 “ ( 𝐵 ∪ 𝐶 ) ) = ( ( 𝐴 “ 𝐵 ) ∪ ( 𝐴 “ 𝐶 ) )

Proof

Step Hyp Ref Expression
1 resundi ⊢ ( 𝐴 ↾ ( 𝐵 ∪ 𝐶 ) ) = ( ( 𝐴 ↾ 𝐵 ) ∪ ( 𝐴 ↾ 𝐶 ) )
2 1 rneqi ⊢ ran ( 𝐴 ↾ ( 𝐵 ∪ 𝐶 ) ) = ran ( ( 𝐴 ↾ 𝐵 ) ∪ ( 𝐴 ↾ 𝐶 ) )
3 rnun ⊢ ran ( ( 𝐴 ↾ 𝐵 ) ∪ ( 𝐴 ↾ 𝐶 ) ) = ( ran ( 𝐴 ↾ 𝐵 ) ∪ ran ( 𝐴 ↾ 𝐶 ) )
4 2 3 eqtri ⊢ ran ( 𝐴 ↾ ( 𝐵 ∪ 𝐶 ) ) = ( ran ( 𝐴 ↾ 𝐵 ) ∪ ran ( 𝐴 ↾ 𝐶 ) )
5 df-ima ⊢ ( 𝐴 “ ( 𝐵 ∪ 𝐶 ) ) = ran ( 𝐴 ↾ ( 𝐵 ∪ 𝐶 ) )
6 df-ima ⊢ ( 𝐴 “ 𝐵 ) = ran ( 𝐴 ↾ 𝐵 )
7 df-ima ⊢ ( 𝐴 “ 𝐶 ) = ran ( 𝐴 ↾ 𝐶 )
8 6 7 uneq12i ⊢ ( ( 𝐴 “ 𝐵 ) ∪ ( 𝐴 “ 𝐶 ) ) = ( ran ( 𝐴 ↾ 𝐵 ) ∪ ran ( 𝐴 ↾ 𝐶 ) )
9 4 5 8 3eqtr4i ⊢ ( 𝐴 “ ( 𝐵 ∪ 𝐶 ) ) = ( ( 𝐴 “ 𝐵 ) ∪ ( 𝐴 “ 𝐶 ) )