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 : 𝐴1-1-onto𝐵𝐶𝐵 ) → ( 𝐹𝐶 ) = ( 𝑥𝐴 ( 𝐹𝑥 ) = 𝐶 ) )

Proof

Step Hyp Ref Expression
1 f1ocnvdm ( ( 𝐹 : 𝐴1-1-onto𝐵𝐶𝐵 ) → ( 𝐹𝐶 ) ∈ 𝐴 )
2 f1ocnvfvb ( ( 𝐹 : 𝐴1-1-onto𝐵𝑥𝐴𝐶𝐵 ) → ( ( 𝐹𝑥 ) = 𝐶 ↔ ( 𝐹𝐶 ) = 𝑥 ) )
3 2 3expa ( ( ( 𝐹 : 𝐴1-1-onto𝐵𝑥𝐴 ) ∧ 𝐶𝐵 ) → ( ( 𝐹𝑥 ) = 𝐶 ↔ ( 𝐹𝐶 ) = 𝑥 ) )
4 3 an32s ( ( ( 𝐹 : 𝐴1-1-onto𝐵𝐶𝐵 ) ∧ 𝑥𝐴 ) → ( ( 𝐹𝑥 ) = 𝐶 ↔ ( 𝐹𝐶 ) = 𝑥 ) )
5 eqcom ( 𝑥 = ( 𝐹𝐶 ) ↔ ( 𝐹𝐶 ) = 𝑥 )
6 4 5 bitr4di ( ( ( 𝐹 : 𝐴1-1-onto𝐵𝐶𝐵 ) ∧ 𝑥𝐴 ) → ( ( 𝐹𝑥 ) = 𝐶𝑥 = ( 𝐹𝐶 ) ) )
7 1 6 riota5 ( ( 𝐹 : 𝐴1-1-onto𝐵𝐶𝐵 ) → ( 𝑥𝐴 ( 𝐹𝑥 ) = 𝐶 ) = ( 𝐹𝐶 ) )
8 7 eqcomd ( ( 𝐹 : 𝐴1-1-onto𝐵𝐶𝐵 ) → ( 𝐹𝐶 ) = ( 𝑥𝐴 ( 𝐹𝑥 ) = 𝐶 ) )