Description: The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng does not require the Axiom of Replacement. (Contributed by NM, 13-Jan-2007) (Revised by Mario Carneiro, 10-Sep-2015)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | f1oen3g | ⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝐹 : 𝐴 –1-1-onto→ 𝐵 ) → 𝐴 ≈ 𝐵 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | f1oeq1 | ⊢ ( 𝑓 = 𝐹 → ( 𝑓 : 𝐴 –1-1-onto→ 𝐵 ↔ 𝐹 : 𝐴 –1-1-onto→ 𝐵 ) ) | |
| 2 | 1 | spcegv | ⊢ ( 𝐹 ∈ 𝑉 → ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 → ∃ 𝑓 𝑓 : 𝐴 –1-1-onto→ 𝐵 ) ) |
| 3 | 2 | imp | ⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝐹 : 𝐴 –1-1-onto→ 𝐵 ) → ∃ 𝑓 𝑓 : 𝐴 –1-1-onto→ 𝐵 ) |
| 4 | bren | ⊢ ( 𝐴 ≈ 𝐵 ↔ ∃ 𝑓 𝑓 : 𝐴 –1-1-onto→ 𝐵 ) | |
| 5 | 3 4 | sylibr | ⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝐹 : 𝐴 –1-1-onto→ 𝐵 ) → 𝐴 ≈ 𝐵 ) |