Description: Equality theorem for one-to-one onto functions. (Contributed by FL, 14-Jul-2012)
Ref | Expression | ||
---|---|---|---|
Assertion | f1oeq23 | ⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ) → ( 𝐹 : 𝐴 –1-1-onto→ 𝐶 ↔ 𝐹 : 𝐵 –1-1-onto→ 𝐷 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1oeq2 | ⊢ ( 𝐴 = 𝐵 → ( 𝐹 : 𝐴 –1-1-onto→ 𝐶 ↔ 𝐹 : 𝐵 –1-1-onto→ 𝐶 ) ) | |
2 | f1oeq3 | ⊢ ( 𝐶 = 𝐷 → ( 𝐹 : 𝐵 –1-1-onto→ 𝐶 ↔ 𝐹 : 𝐵 –1-1-onto→ 𝐷 ) ) | |
3 | 1 2 | sylan9bb | ⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ) → ( 𝐹 : 𝐴 –1-1-onto→ 𝐶 ↔ 𝐹 : 𝐵 –1-1-onto→ 𝐷 ) ) |