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 ) | |
f1od.2 | ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐶 ∈ 𝑊 ) | ||
f1od.3 | ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → 𝐷 ∈ 𝑋 ) | ||
f1od.4 | ⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶 ) ↔ ( 𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷 ) ) ) | ||
Assertion | f1od | ⊢ ( 𝜑 → 𝐹 : 𝐴 –1-1-onto→ 𝐵 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1od.1 | ⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) | |
2 | f1od.2 | ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐶 ∈ 𝑊 ) | |
3 | f1od.3 | ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → 𝐷 ∈ 𝑋 ) | |
4 | f1od.4 | ⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶 ) ↔ ( 𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷 ) ) ) | |
5 | 1 2 3 4 | f1ocnvd | ⊢ ( 𝜑 → ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ ◡ 𝐹 = ( 𝑦 ∈ 𝐵 ↦ 𝐷 ) ) ) |
6 | 5 | simpld | ⊢ ( 𝜑 → 𝐹 : 𝐴 –1-1-onto→ 𝐵 ) |