Metamath Proof Explorer


Theorem f1o2d

Description: Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 12-May-2014)

Ref Expression
Hypotheses f1od.1 𝐹 = ( 𝑥𝐴𝐶 )
f1o2d.2 ( ( 𝜑𝑥𝐴 ) → 𝐶𝐵 )
f1o2d.3 ( ( 𝜑𝑦𝐵 ) → 𝐷𝐴 )
f1o2d.4 ( ( 𝜑 ∧ ( 𝑥𝐴𝑦𝐵 ) ) → ( 𝑥 = 𝐷𝑦 = 𝐶 ) )
Assertion f1o2d ( 𝜑𝐹 : 𝐴1-1-onto𝐵 )

Proof

Step Hyp Ref Expression
1 f1od.1 𝐹 = ( 𝑥𝐴𝐶 )
2 f1o2d.2 ( ( 𝜑𝑥𝐴 ) → 𝐶𝐵 )
3 f1o2d.3 ( ( 𝜑𝑦𝐵 ) → 𝐷𝐴 )
4 f1o2d.4 ( ( 𝜑 ∧ ( 𝑥𝐴𝑦𝐵 ) ) → ( 𝑥 = 𝐷𝑦 = 𝐶 ) )
5 1 2 3 4 f1ocnv2d ( 𝜑 → ( 𝐹 : 𝐴1-1-onto𝐵 𝐹 = ( 𝑦𝐵𝐷 ) ) )
6 5 simpld ( 𝜑𝐹 : 𝐴1-1-onto𝐵 )