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Metamath Proof Explorer

Theorem fvtp3g

Description: The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017)

Ref Expression
Assertion fvtp3g ( ( ( 𝐶 ∈ 𝑉 ∧ 𝐹 ∈ 𝑊 ) ∧ ( 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ) ) → ( { ⟨ 𝐴 , 𝐷 ⟩ , ⟨ 𝐵 , 𝐸 ⟩ , ⟨ 𝐶 , 𝐹 ⟩ } ‘ 𝐶 ) = 𝐹 )

Proof

Step Hyp Ref Expression
1 tprot ⊢ { ⟨ 𝐴 , 𝐷 ⟩ , ⟨ 𝐵 , 𝐸 ⟩ , ⟨ 𝐶 , 𝐹 ⟩ } = { ⟨ 𝐵 , 𝐸 ⟩ , ⟨ 𝐶 , 𝐹 ⟩ , ⟨ 𝐴 , 𝐷 ⟩ }
2 1 fveq1i ⊢ ( { ⟨ 𝐴 , 𝐷 ⟩ , ⟨ 𝐵 , 𝐸 ⟩ , ⟨ 𝐶 , 𝐹 ⟩ } ‘ 𝐶 ) = ( { ⟨ 𝐵 , 𝐸 ⟩ , ⟨ 𝐶 , 𝐹 ⟩ , ⟨ 𝐴 , 𝐷 ⟩ } ‘ 𝐶 )
3 necom ⊢ ( 𝐴 ≠ 𝐶 ↔ 𝐶 ≠ 𝐴 )
4 fvtp2g ⊢ ( ( ( 𝐶 ∈ 𝑉 ∧ 𝐹 ∈ 𝑊 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴 ) ) → ( { ⟨ 𝐵 , 𝐸 ⟩ , ⟨ 𝐶 , 𝐹 ⟩ , ⟨ 𝐴 , 𝐷 ⟩ } ‘ 𝐶 ) = 𝐹 )
5 4 expcom ⊢ ( ( 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴 ) → ( ( 𝐶 ∈ 𝑉 ∧ 𝐹 ∈ 𝑊 ) → ( { ⟨ 𝐵 , 𝐸 ⟩ , ⟨ 𝐶 , 𝐹 ⟩ , ⟨ 𝐴 , 𝐷 ⟩ } ‘ 𝐶 ) = 𝐹 ) )
6 3 5 sylan2b ⊢ ( ( 𝐵 ≠ 𝐶 ∧ 𝐴 ≠ 𝐶 ) → ( ( 𝐶 ∈ 𝑉 ∧ 𝐹 ∈ 𝑊 ) → ( { ⟨ 𝐵 , 𝐸 ⟩ , ⟨ 𝐶 , 𝐹 ⟩ , ⟨ 𝐴 , 𝐷 ⟩ } ‘ 𝐶 ) = 𝐹 ) )
7 6 ancoms ⊢ ( ( 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ) → ( ( 𝐶 ∈ 𝑉 ∧ 𝐹 ∈ 𝑊 ) → ( { ⟨ 𝐵 , 𝐸 ⟩ , ⟨ 𝐶 , 𝐹 ⟩ , ⟨ 𝐴 , 𝐷 ⟩ } ‘ 𝐶 ) = 𝐹 ) )
8 7 impcom ⊢ ( ( ( 𝐶 ∈ 𝑉 ∧ 𝐹 ∈ 𝑊 ) ∧ ( 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ) ) → ( { ⟨ 𝐵 , 𝐸 ⟩ , ⟨ 𝐶 , 𝐹 ⟩ , ⟨ 𝐴 , 𝐷 ⟩ } ‘ 𝐶 ) = 𝐹 )
9 2 8 eqtrid ⊢ ( ( ( 𝐶 ∈ 𝑉 ∧ 𝐹 ∈ 𝑊 ) ∧ ( 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ) ) → ( { ⟨ 𝐴 , 𝐷 ⟩ , ⟨ 𝐵 , 𝐸 ⟩ , ⟨ 𝐶 , 𝐹 ⟩ } ‘ 𝐶 ) = 𝐹 )