Description: Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 12-May-2014)
Ref | Expression | ||
---|---|---|---|
Hypotheses | f1od.1 | ⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) | |
f1o2d.2 | ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐶 ∈ 𝐵 ) | ||
f1o2d.3 | ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → 𝐷 ∈ 𝐴 ) | ||
f1o2d.4 | ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 = 𝐷 ↔ 𝑦 = 𝐶 ) ) | ||
Assertion | f1o2d | ⊢ ( 𝜑 → 𝐹 : 𝐴 –1-1-onto→ 𝐵 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1od.1 | ⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) | |
2 | f1o2d.2 | ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐶 ∈ 𝐵 ) | |
3 | f1o2d.3 | ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → 𝐷 ∈ 𝐴 ) | |
4 | f1o2d.4 | ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 = 𝐷 ↔ 𝑦 = 𝐶 ) ) | |
5 | 1 2 3 4 | f1ocnv2d | ⊢ ( 𝜑 → ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ ◡ 𝐹 = ( 𝑦 ∈ 𝐵 ↦ 𝐷 ) ) ) |
6 | 5 | simpld | ⊢ ( 𝜑 → 𝐹 : 𝐴 –1-1-onto→ 𝐵 ) |