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 ) } )