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Metamath Proof Explorer

Theorem f1ocnvfv3

Description: Value of the converse of a one-to-one onto function. (Contributed by NM, 26-May-2006) (Proof shortened by Mario Carneiro, 24-Dec-2016)

Ref Expression
Assertion f1ocnvfv3 ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐵 ) → ( ◡ 𝐹 ‘ 𝐶 ) = ( ℩ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝐶 ) )

Proof

Step Hyp Ref Expression
1 f1ocnvdm ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐵 ) → ( ◡ 𝐹 ‘ 𝐶 ) ∈ 𝐴 )
2 f1ocnvfvb ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵 ) → ( ( 𝐹 ‘ 𝑥 ) = 𝐶 ↔ ( ◡ 𝐹 ‘ 𝐶 ) = 𝑥 ) )
3 2 3expa ⊢ ( ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝐶 ∈ 𝐵 ) → ( ( 𝐹 ‘ 𝑥 ) = 𝐶 ↔ ( ◡ 𝐹 ‘ 𝐶 ) = 𝑥 ) )
4 3 an32s ⊢ ( ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝐹 ‘ 𝑥 ) = 𝐶 ↔ ( ◡ 𝐹 ‘ 𝐶 ) = 𝑥 ) )
5 eqcom ⊢ ( 𝑥 = ( ◡ 𝐹 ‘ 𝐶 ) ↔ ( ◡ 𝐹 ‘ 𝐶 ) = 𝑥 )
6 4 5 bitr4di ⊢ ( ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝐹 ‘ 𝑥 ) = 𝐶 ↔ 𝑥 = ( ◡ 𝐹 ‘ 𝐶 ) ) )
7 1 6 riota5 ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐵 ) → ( ℩ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝐶 ) = ( ◡ 𝐹 ‘ 𝐶 ) )
8 7 eqcomd ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐵 ) → ( ◡ 𝐹 ‘ 𝐶 ) = ( ℩ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝐶 ) )