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Metamath Proof Explorer

Theorem fvtp1

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

Ref Expression
Hypotheses fvtp1.1 ⊢ 𝐴 ∈ V
fvtp1.4 ⊢ 𝐷 ∈ V
Assertion fvtp1 ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) → ( { ⟨ 𝐴 , 𝐷 ⟩ , ⟨ 𝐵 , 𝐸 ⟩ , ⟨ 𝐶 , 𝐹 ⟩ } ‘ 𝐴 ) = 𝐷 )

Proof

Step Hyp Ref Expression
1 fvtp1.1 ⊢ 𝐴 ∈ V
2 fvtp1.4 ⊢ 𝐷 ∈ V
3 df-tp ⊢ { ⟨ 𝐴 , 𝐷 ⟩ , ⟨ 𝐵 , 𝐸 ⟩ , ⟨ 𝐶 , 𝐹 ⟩ } = ( { ⟨ 𝐴 , 𝐷 ⟩ , ⟨ 𝐵 , 𝐸 ⟩ } ∪ { ⟨ 𝐶 , 𝐹 ⟩ } )
4 3 fveq1i ⊢ ( { ⟨ 𝐴 , 𝐷 ⟩ , ⟨ 𝐵 , 𝐸 ⟩ , ⟨ 𝐶 , 𝐹 ⟩ } ‘ 𝐴 ) = ( ( { ⟨ 𝐴 , 𝐷 ⟩ , ⟨ 𝐵 , 𝐸 ⟩ } ∪ { ⟨ 𝐶 , 𝐹 ⟩ } ) ‘ 𝐴 )
5 necom ⊢ ( 𝐴 ≠ 𝐶 ↔ 𝐶 ≠ 𝐴 )
6 fvunsn ⊢ ( 𝐶 ≠ 𝐴 → ( ( { ⟨ 𝐴 , 𝐷 ⟩ , ⟨ 𝐵 , 𝐸 ⟩ } ∪ { ⟨ 𝐶 , 𝐹 ⟩ } ) ‘ 𝐴 ) = ( { ⟨ 𝐴 , 𝐷 ⟩ , ⟨ 𝐵 , 𝐸 ⟩ } ‘ 𝐴 ) )
7 5 6 sylbi ⊢ ( 𝐴 ≠ 𝐶 → ( ( { ⟨ 𝐴 , 𝐷 ⟩ , ⟨ 𝐵 , 𝐸 ⟩ } ∪ { ⟨ 𝐶 , 𝐹 ⟩ } ) ‘ 𝐴 ) = ( { ⟨ 𝐴 , 𝐷 ⟩ , ⟨ 𝐵 , 𝐸 ⟩ } ‘ 𝐴 ) )
8 1 2 fvpr1 ⊢ ( 𝐴 ≠ 𝐵 → ( { ⟨ 𝐴 , 𝐷 ⟩ , ⟨ 𝐵 , 𝐸 ⟩ } ‘ 𝐴 ) = 𝐷 )
9 7 8 sylan9eqr ⊢ ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) → ( ( { ⟨ 𝐴 , 𝐷 ⟩ , ⟨ 𝐵 , 𝐸 ⟩ } ∪ { ⟨ 𝐶 , 𝐹 ⟩ } ) ‘ 𝐴 ) = 𝐷 )
10 4 9 eqtrid ⊢ ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) → ( { ⟨ 𝐴 , 𝐷 ⟩ , ⟨ 𝐵 , 𝐸 ⟩ , ⟨ 𝐶 , 𝐹 ⟩ } ‘ 𝐴 ) = 𝐷 )