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Metamath Proof Explorer

Theorem fvtp3

Description: The third value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011)

Ref Expression
Hypotheses fvtp3.1 ⊢ 𝐶 ∈ V
fvtp3.4 ⊢ 𝐹 ∈ V
Assertion fvtp3 ( ( 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ) → ( { ⟨ 𝐴 , 𝐷 ⟩ , ⟨ 𝐵 , 𝐸 ⟩ , ⟨ 𝐶 , 𝐹 ⟩ } ‘ 𝐶 ) = 𝐹 )

Proof

Step Hyp Ref Expression
1 fvtp3.1 ⊢ 𝐶 ∈ V
2 fvtp3.4 ⊢ 𝐹 ∈ V
3 tprot ⊢ { ⟨ 𝐴 , 𝐷 ⟩ , ⟨ 𝐵 , 𝐸 ⟩ , ⟨ 𝐶 , 𝐹 ⟩ } = { ⟨ 𝐵 , 𝐸 ⟩ , ⟨ 𝐶 , 𝐹 ⟩ , ⟨ 𝐴 , 𝐷 ⟩ }
4 3 fveq1i ⊢ ( { ⟨ 𝐴 , 𝐷 ⟩ , ⟨ 𝐵 , 𝐸 ⟩ , ⟨ 𝐶 , 𝐹 ⟩ } ‘ 𝐶 ) = ( { ⟨ 𝐵 , 𝐸 ⟩ , ⟨ 𝐶 , 𝐹 ⟩ , ⟨ 𝐴 , 𝐷 ⟩ } ‘ 𝐶 )
5 necom ⊢ ( 𝐴 ≠ 𝐶 ↔ 𝐶 ≠ 𝐴 )
6 1 2 fvtp2 ⊢ ( ( 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴 ) → ( { ⟨ 𝐵 , 𝐸 ⟩ , ⟨ 𝐶 , 𝐹 ⟩ , ⟨ 𝐴 , 𝐷 ⟩ } ‘ 𝐶 ) = 𝐹 )
7 5 6 sylan2b ⊢ ( ( 𝐵 ≠ 𝐶 ∧ 𝐴 ≠ 𝐶 ) → ( { ⟨ 𝐵 , 𝐸 ⟩ , ⟨ 𝐶 , 𝐹 ⟩ , ⟨ 𝐴 , 𝐷 ⟩ } ‘ 𝐶 ) = 𝐹 )
8 7 ancoms ⊢ ( ( 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ) → ( { ⟨ 𝐵 , 𝐸 ⟩ , ⟨ 𝐶 , 𝐹 ⟩ , ⟨ 𝐴 , 𝐷 ⟩ } ‘ 𝐶 ) = 𝐹 )
9 4 8 eqtrid ⊢ ( ( 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ) → ( { ⟨ 𝐴 , 𝐷 ⟩ , ⟨ 𝐵 , 𝐸 ⟩ , ⟨ 𝐶 , 𝐹 ⟩ } ‘ 𝐶 ) = 𝐹 )