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Metamath Proof Explorer

Theorem fnimapr

Description: The image of a pair under a function. (Contributed by Jeff Madsen, 6-Jan-2011)

Ref Expression
Assertion fnimapr ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ) → ( 𝐹 “ { 𝐵 , 𝐶 } ) = { ( 𝐹 ‘ 𝐵 ) , ( 𝐹 ‘ 𝐶 ) } )

Proof

Step Hyp Ref Expression
1 fnsnfv ⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ) → { ( 𝐹 ‘ 𝐵 ) } = ( 𝐹 “ { 𝐵 } ) )
2 1 3adant3 ⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ) → { ( 𝐹 ‘ 𝐵 ) } = ( 𝐹 “ { 𝐵 } ) )
3 fnsnfv ⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐶 ∈ 𝐴 ) → { ( 𝐹 ‘ 𝐶 ) } = ( 𝐹 “ { 𝐶 } ) )
4 3 3adant2 ⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ) → { ( 𝐹 ‘ 𝐶 ) } = ( 𝐹 “ { 𝐶 } ) )
5 2 4 uneq12d ⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ) → ( { ( 𝐹 ‘ 𝐵 ) } ∪ { ( 𝐹 ‘ 𝐶 ) } ) = ( ( 𝐹 “ { 𝐵 } ) ∪ ( 𝐹 “ { 𝐶 } ) ) )
6 5 eqcomd ⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ) → ( ( 𝐹 “ { 𝐵 } ) ∪ ( 𝐹 “ { 𝐶 } ) ) = ( { ( 𝐹 ‘ 𝐵 ) } ∪ { ( 𝐹 ‘ 𝐶 ) } ) )
7 df-pr ⊢ { 𝐵 , 𝐶 } = ( { 𝐵 } ∪ { 𝐶 } )
8 7 imaeq2i ⊢ ( 𝐹 “ { 𝐵 , 𝐶 } ) = ( 𝐹 “ ( { 𝐵 } ∪ { 𝐶 } ) )
9 imaundi ⊢ ( 𝐹 “ ( { 𝐵 } ∪ { 𝐶 } ) ) = ( ( 𝐹 “ { 𝐵 } ) ∪ ( 𝐹 “ { 𝐶 } ) )
10 8 9 eqtri ⊢ ( 𝐹 “ { 𝐵 , 𝐶 } ) = ( ( 𝐹 “ { 𝐵 } ) ∪ ( 𝐹 “ { 𝐶 } ) )
11 df-pr ⊢ { ( 𝐹 ‘ 𝐵 ) , ( 𝐹 ‘ 𝐶 ) } = ( { ( 𝐹 ‘ 𝐵 ) } ∪ { ( 𝐹 ‘ 𝐶 ) } )
12 6 10 11 3eqtr4g ⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ) → ( 𝐹 “ { 𝐵 , 𝐶 } ) = { ( 𝐹 ‘ 𝐵 ) , ( 𝐹 ‘ 𝐶 ) } )