Description: The domain of a one-to-one set function is dominated by its codomain when the latter is a set. This variation of f1domg does not require the Axiom of Replacement nor the Axiom of Power Sets. (Contributed by BTernaryTau, 9-Sep-2024)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | f1dom3g | ⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹 : 𝐴 –1-1→ 𝐵 ) → 𝐴 ≼ 𝐵 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | f1eq1 | ⊢ ( 𝑓 = 𝐹 → ( 𝑓 : 𝐴 –1-1→ 𝐵 ↔ 𝐹 : 𝐴 –1-1→ 𝐵 ) ) | |
| 2 | 1 | spcegv | ⊢ ( 𝐹 ∈ 𝑉 → ( 𝐹 : 𝐴 –1-1→ 𝐵 → ∃ 𝑓 𝑓 : 𝐴 –1-1→ 𝐵 ) ) |
| 3 | 2 | imp | ⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝐹 : 𝐴 –1-1→ 𝐵 ) → ∃ 𝑓 𝑓 : 𝐴 –1-1→ 𝐵 ) |
| 4 | 3 | 3adant2 | ⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹 : 𝐴 –1-1→ 𝐵 ) → ∃ 𝑓 𝑓 : 𝐴 –1-1→ 𝐵 ) |
| 5 | brdomg | ⊢ ( 𝐵 ∈ 𝑊 → ( 𝐴 ≼ 𝐵 ↔ ∃ 𝑓 𝑓 : 𝐴 –1-1→ 𝐵 ) ) | |
| 6 | 5 | 3ad2ant2 | ⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹 : 𝐴 –1-1→ 𝐵 ) → ( 𝐴 ≼ 𝐵 ↔ ∃ 𝑓 𝑓 : 𝐴 –1-1→ 𝐵 ) ) |
| 7 | 4 6 | mpbird | ⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹 : 𝐴 –1-1→ 𝐵 ) → 𝐴 ≼ 𝐵 ) |