Metamath Proof Explorer


Theorem f1resrcmplf1d

Description: If a function's restriction to a subclass of its domain and its restriction to the relative complement of that subclass are both one-to-one, and if the ranges of those two restrictions are disjoint, then the function is itself one-to-one. (Contributed by BTernaryTau, 28-Sep-2023)

Ref Expression
Hypotheses f1resrcmplf1d.1 ( 𝜑𝐶𝐴 )
f1resrcmplf1d.2 ( 𝜑𝐹 : 𝐴𝐵 )
f1resrcmplf1d.3 ( 𝜑 → ( 𝐹𝐶 ) : 𝐶1-1𝐵 )
f1resrcmplf1d.4 ( 𝜑 → ( 𝐹 ↾ ( 𝐴𝐶 ) ) : ( 𝐴𝐶 ) –1-1𝐵 )
f1resrcmplf1d.5 ( 𝜑 → ( ( 𝐹𝐶 ) ∩ ( 𝐹 “ ( 𝐴𝐶 ) ) ) = ∅ )
Assertion f1resrcmplf1d ( 𝜑𝐹 : 𝐴1-1𝐵 )

Proof

Step Hyp Ref Expression
1 f1resrcmplf1d.1 ( 𝜑𝐶𝐴 )
2 f1resrcmplf1d.2 ( 𝜑𝐹 : 𝐴𝐵 )
3 f1resrcmplf1d.3 ( 𝜑 → ( 𝐹𝐶 ) : 𝐶1-1𝐵 )
4 f1resrcmplf1d.4 ( 𝜑 → ( 𝐹 ↾ ( 𝐴𝐶 ) ) : ( 𝐴𝐶 ) –1-1𝐵 )
5 f1resrcmplf1d.5 ( 𝜑 → ( ( 𝐹𝐶 ) ∩ ( 𝐹 “ ( 𝐴𝐶 ) ) ) = ∅ )
6 f1resveqaeq ( ( ( 𝐹𝐶 ) : 𝐶1-1𝐵 ∧ ( 𝑥𝐶𝑦𝐶 ) ) → ( ( 𝐹𝑥 ) = ( 𝐹𝑦 ) → 𝑥 = 𝑦 ) )
7 3 6 sylan ( ( 𝜑 ∧ ( 𝑥𝐶𝑦𝐶 ) ) → ( ( 𝐹𝑥 ) = ( 𝐹𝑦 ) → 𝑥 = 𝑦 ) )
8 7 ex ( 𝜑 → ( ( 𝑥𝐶𝑦𝐶 ) → ( ( 𝐹𝑥 ) = ( 𝐹𝑦 ) → 𝑥 = 𝑦 ) ) )
9 difssd ( 𝜑 → ( 𝐴𝐶 ) ⊆ 𝐴 )
10 1 9 2 5 f1resrcmplf1dlem ( 𝜑 → ( ( 𝑥𝐶𝑦 ∈ ( 𝐴𝐶 ) ) → ( ( 𝐹𝑥 ) = ( 𝐹𝑦 ) → 𝑥 = 𝑦 ) ) )
11 incom ( ( 𝐹𝐶 ) ∩ ( 𝐹 “ ( 𝐴𝐶 ) ) ) = ( ( 𝐹 “ ( 𝐴𝐶 ) ) ∩ ( 𝐹𝐶 ) )
12 11 5 eqtr3id ( 𝜑 → ( ( 𝐹 “ ( 𝐴𝐶 ) ) ∩ ( 𝐹𝐶 ) ) = ∅ )
13 9 1 2 12 f1resrcmplf1dlem ( 𝜑 → ( ( 𝑥 ∈ ( 𝐴𝐶 ) ∧ 𝑦𝐶 ) → ( ( 𝐹𝑥 ) = ( 𝐹𝑦 ) → 𝑥 = 𝑦 ) ) )
14 f1resveqaeq ( ( ( 𝐹 ↾ ( 𝐴𝐶 ) ) : ( 𝐴𝐶 ) –1-1𝐵 ∧ ( 𝑥 ∈ ( 𝐴𝐶 ) ∧ 𝑦 ∈ ( 𝐴𝐶 ) ) ) → ( ( 𝐹𝑥 ) = ( 𝐹𝑦 ) → 𝑥 = 𝑦 ) )
15 4 14 sylan ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝐴𝐶 ) ∧ 𝑦 ∈ ( 𝐴𝐶 ) ) ) → ( ( 𝐹𝑥 ) = ( 𝐹𝑦 ) → 𝑥 = 𝑦 ) )
16 15 ex ( 𝜑 → ( ( 𝑥 ∈ ( 𝐴𝐶 ) ∧ 𝑦 ∈ ( 𝐴𝐶 ) ) → ( ( 𝐹𝑥 ) = ( 𝐹𝑦 ) → 𝑥 = 𝑦 ) ) )
17 8 10 13 16 prsrcmpltd ( 𝜑 → ( ( 𝑥𝐴𝑦𝐴 ) → ( ( 𝐹𝑥 ) = ( 𝐹𝑦 ) → 𝑥 = 𝑦 ) ) )
18 17 ralrimivv ( 𝜑 → ∀ 𝑥𝐴𝑦𝐴 ( ( 𝐹𝑥 ) = ( 𝐹𝑦 ) → 𝑥 = 𝑦 ) )
19 dff13 ( 𝐹 : 𝐴1-1𝐵 ↔ ( 𝐹 : 𝐴𝐵 ∧ ∀ 𝑥𝐴𝑦𝐴 ( ( 𝐹𝑥 ) = ( 𝐹𝑦 ) → 𝑥 = 𝑦 ) ) )
20 2 18 19 sylanbrc ( 𝜑𝐹 : 𝐴1-1𝐵 )