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Metamath Proof Explorer

Theorem inimass

Description: The image of an intersection. (Contributed by Thierry Arnoux, 16-Dec-2017)

Ref Expression
Assertion inimass ( ( 𝐴 ∩ 𝐵 ) “ 𝐶 ) ⊆ ( ( 𝐴 “ 𝐶 ) ∩ ( 𝐵 “ 𝐶 ) )

Proof

Step Hyp Ref Expression
1 rnin ⊢ ran ( ( 𝐴 ↾ 𝐶 ) ∩ ( 𝐵 ↾ 𝐶 ) ) ⊆ ( ran ( 𝐴 ↾ 𝐶 ) ∩ ran ( 𝐵 ↾ 𝐶 ) )
2 df-ima ⊢ ( ( 𝐴 ∩ 𝐵 ) “ 𝐶 ) = ran ( ( 𝐴 ∩ 𝐵 ) ↾ 𝐶 )
3 resindir ⊢ ( ( 𝐴 ∩ 𝐵 ) ↾ 𝐶 ) = ( ( 𝐴 ↾ 𝐶 ) ∩ ( 𝐵 ↾ 𝐶 ) )
4 3 rneqi ⊢ ran ( ( 𝐴 ∩ 𝐵 ) ↾ 𝐶 ) = ran ( ( 𝐴 ↾ 𝐶 ) ∩ ( 𝐵 ↾ 𝐶 ) )
5 2 4 eqtri ⊢ ( ( 𝐴 ∩ 𝐵 ) “ 𝐶 ) = ran ( ( 𝐴 ↾ 𝐶 ) ∩ ( 𝐵 ↾ 𝐶 ) )
6 df-ima ⊢ ( 𝐴 “ 𝐶 ) = ran ( 𝐴 ↾ 𝐶 )
7 df-ima ⊢ ( 𝐵 “ 𝐶 ) = ran ( 𝐵 ↾ 𝐶 )
8 6 7 ineq12i ⊢ ( ( 𝐴 “ 𝐶 ) ∩ ( 𝐵 “ 𝐶 ) ) = ( ran ( 𝐴 ↾ 𝐶 ) ∩ ran ( 𝐵 ↾ 𝐶 ) )
9 1 5 8 3sstr4i ⊢ ( ( 𝐴 ∩ 𝐵 ) “ 𝐶 ) ⊆ ( ( 𝐴 “ 𝐶 ) ∩ ( 𝐵 “ 𝐶 ) )