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Metamath Proof Explorer

Theorem f1ocnvfv1

Description: The converse value of the value of a one-to-one onto function. (Contributed by NM, 20-May-2004)

Ref Expression
Assertion f1ocnvfv1 ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐴 ) → ( ◡ 𝐹 ‘ ( 𝐹 ‘ 𝐶 ) ) = 𝐶 )

Proof

Step Hyp Ref Expression
1 f1ococnv1 ⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 → ( ◡ 𝐹 ∘ 𝐹 ) = ( I ↾ 𝐴 ) )
2 1 fveq1d ⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 → ( ( ◡ 𝐹 ∘ 𝐹 ) ‘ 𝐶 ) = ( ( I ↾ 𝐴 ) ‘ 𝐶 ) )
3 2 adantr ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐴 ) → ( ( ◡ 𝐹 ∘ 𝐹 ) ‘ 𝐶 ) = ( ( I ↾ 𝐴 ) ‘ 𝐶 ) )
4 f1of ⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 → 𝐹 : 𝐴 ⟶ 𝐵 )
5 fvco3 ⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐶 ∈ 𝐴 ) → ( ( ◡ 𝐹 ∘ 𝐹 ) ‘ 𝐶 ) = ( ◡ 𝐹 ‘ ( 𝐹 ‘ 𝐶 ) ) )
6 4 5 sylan ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐴 ) → ( ( ◡ 𝐹 ∘ 𝐹 ) ‘ 𝐶 ) = ( ◡ 𝐹 ‘ ( 𝐹 ‘ 𝐶 ) ) )
7 fvresi ⊢ ( 𝐶 ∈ 𝐴 → ( ( I ↾ 𝐴 ) ‘ 𝐶 ) = 𝐶 )
8 7 adantl ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐴 ) → ( ( I ↾ 𝐴 ) ‘ 𝐶 ) = 𝐶 )
9 3 6 8 3eqtr3d ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐴 ) → ( ◡ 𝐹 ‘ ( 𝐹 ‘ 𝐶 ) ) = 𝐶 )