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Metamath Proof Explorer

Theorem f1domg

Description: The domain of a one-to-one function is dominated by its codomain. (Contributed by NM, 4-Sep-2004)

Ref Expression
Assertion f1domg ( 𝐵 ∈ 𝐶 → ( 𝐹 : 𝐴 –1-1→ 𝐵 → 𝐴 ≼ 𝐵 ) )

Proof

Step Hyp Ref Expression
1 f1f ⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 → 𝐹 : 𝐴 ⟶ 𝐵 )
2 f1dmex ⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ 𝐵 ∈ 𝐶 ) → 𝐴 ∈ V )
3 fex ⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐴 ∈ V ) → 𝐹 ∈ V )
4 1 2 3 syl2an2r ⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ 𝐵 ∈ 𝐶 ) → 𝐹 ∈ V )
5 4 expcom ⊢ ( 𝐵 ∈ 𝐶 → ( 𝐹 : 𝐴 –1-1→ 𝐵 → 𝐹 ∈ V ) )
6 f1eq1 ⊢ ( 𝑓 = 𝐹 → ( 𝑓 : 𝐴 –1-1→ 𝐵 ↔ 𝐹 : 𝐴 –1-1→ 𝐵 ) )
7 6 spcegv ⊢ ( 𝐹 ∈ V → ( 𝐹 : 𝐴 –1-1→ 𝐵 → ∃ 𝑓 𝑓 : 𝐴 –1-1→ 𝐵 ) )
8 5 7 syli ⊢ ( 𝐵 ∈ 𝐶 → ( 𝐹 : 𝐴 –1-1→ 𝐵 → ∃ 𝑓 𝑓 : 𝐴 –1-1→ 𝐵 ) )
9 brdomg ⊢ ( 𝐵 ∈ 𝐶 → ( 𝐴 ≼ 𝐵 ↔ ∃ 𝑓 𝑓 : 𝐴 –1-1→ 𝐵 ) )
10 8 9 sylibrd ⊢ ( 𝐵 ∈ 𝐶 → ( 𝐹 : 𝐴 –1-1→ 𝐵 → 𝐴 ≼ 𝐵 ) )