Metamath Proof Explorer
Description: Equality deduction for one-to-one onto functions. (Contributed by Glauco Siliprandi, 17-Aug-2020)
|
|
Ref |
Expression |
|
Hypothesis |
f1oeq1d.1 |
⊢ ( 𝜑 → 𝐹 = 𝐺 ) |
|
Assertion |
f1oeq1d |
⊢ ( 𝜑 → ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ↔ 𝐺 : 𝐴 –1-1-onto→ 𝐵 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
f1oeq1d.1 |
⊢ ( 𝜑 → 𝐹 = 𝐺 ) |
2 |
|
f1oeq1 |
⊢ ( 𝐹 = 𝐺 → ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ↔ 𝐺 : 𝐴 –1-1-onto→ 𝐵 ) ) |
3 |
1 2
|
syl |
⊢ ( 𝜑 → ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ↔ 𝐺 : 𝐴 –1-1-onto→ 𝐵 ) ) |