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Metamath Proof Explorer

Theorem fnimatpd

Description: The image of an unordered triple under a function. (Contributed by Thierry Arnoux, 19-Sep-2023)

Ref Expression
Hypotheses fnimatpd.1 ⊢ ( 𝜑 → 𝐹 Fn 𝐷 )
fnimatpd.2 ⊢ ( 𝜑 → 𝐴 ∈ 𝐷 )
fnimatpd.3 ⊢ ( 𝜑 → 𝐵 ∈ 𝐷 )
fnimatpd.4 ⊢ ( 𝜑 → 𝐶 ∈ 𝐷 )
Assertion fnimatpd ( 𝜑 → ( 𝐹 “ { 𝐴 , 𝐵 , 𝐶 } ) = { ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) , ( 𝐹 ‘ 𝐶 ) } )

Proof

Step Hyp Ref Expression
1 fnimatpd.1 ⊢ ( 𝜑 → 𝐹 Fn 𝐷 )
2 fnimatpd.2 ⊢ ( 𝜑 → 𝐴 ∈ 𝐷 )
3 fnimatpd.3 ⊢ ( 𝜑 → 𝐵 ∈ 𝐷 )
4 fnimatpd.4 ⊢ ( 𝜑 → 𝐶 ∈ 𝐷 )
5 fnimapr ⊢ ( ( 𝐹 Fn 𝐷 ∧ 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ) → ( 𝐹 “ { 𝐴 , 𝐵 } ) = { ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) } )
6 1 2 3 5 syl3anc ⊢ ( 𝜑 → ( 𝐹 “ { 𝐴 , 𝐵 } ) = { ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) } )
7 fnsnfv ⊢ ( ( 𝐹 Fn 𝐷 ∧ 𝐶 ∈ 𝐷 ) → { ( 𝐹 ‘ 𝐶 ) } = ( 𝐹 “ { 𝐶 } ) )
8 1 4 7 syl2anc ⊢ ( 𝜑 → { ( 𝐹 ‘ 𝐶 ) } = ( 𝐹 “ { 𝐶 } ) )
9 8 eqcomd ⊢ ( 𝜑 → ( 𝐹 “ { 𝐶 } ) = { ( 𝐹 ‘ 𝐶 ) } )
10 6 9 uneq12d ⊢ ( 𝜑 → ( ( 𝐹 “ { 𝐴 , 𝐵 } ) ∪ ( 𝐹 “ { 𝐶 } ) ) = ( { ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) } ∪ { ( 𝐹 ‘ 𝐶 ) } ) )
11 df-tp ⊢ { 𝐴 , 𝐵 , 𝐶 } = ( { 𝐴 , 𝐵 } ∪ { 𝐶 } )
12 11 imaeq2i ⊢ ( 𝐹 “ { 𝐴 , 𝐵 , 𝐶 } ) = ( 𝐹 “ ( { 𝐴 , 𝐵 } ∪ { 𝐶 } ) )
13 imaundi ⊢ ( 𝐹 “ ( { 𝐴 , 𝐵 } ∪ { 𝐶 } ) ) = ( ( 𝐹 “ { 𝐴 , 𝐵 } ) ∪ ( 𝐹 “ { 𝐶 } ) )
14 12 13 eqtri ⊢ ( 𝐹 “ { 𝐴 , 𝐵 , 𝐶 } ) = ( ( 𝐹 “ { 𝐴 , 𝐵 } ) ∪ ( 𝐹 “ { 𝐶 } ) )
15 df-tp ⊢ { ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) , ( 𝐹 ‘ 𝐶 ) } = ( { ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) } ∪ { ( 𝐹 ‘ 𝐶 ) } )
16 10 14 15 3eqtr4g ⊢ ( 𝜑 → ( 𝐹 “ { 𝐴 , 𝐵 , 𝐶 } ) = { ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) , ( 𝐹 ‘ 𝐶 ) } )