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Metamath Proof Explorer
Description: Equality deduction for one-to-one onto functions. (Contributed by Glauco Siliprandi, 17-Aug-2020)
|
|
Ref |
Expression |
|
Hypothesis |
f1oeq2d.1 |
⊢ ( 𝜑 → 𝐴 = 𝐵 ) |
|
Assertion |
f1oeq2d |
⊢ ( 𝜑 → ( 𝐹 : 𝐴 –1-1-onto→ 𝐶 ↔ 𝐹 : 𝐵 –1-1-onto→ 𝐶 ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
f1oeq2d.1 |
⊢ ( 𝜑 → 𝐴 = 𝐵 ) |
| 2 |
|
f1oeq2 |
⊢ ( 𝐴 = 𝐵 → ( 𝐹 : 𝐴 –1-1-onto→ 𝐶 ↔ 𝐹 : 𝐵 –1-1-onto→ 𝐶 ) ) |
| 3 |
1 2
|
syl |
⊢ ( 𝜑 → ( 𝐹 : 𝐴 –1-1-onto→ 𝐶 ↔ 𝐹 : 𝐵 –1-1-onto→ 𝐶 ) ) |