Theorem ffrnbd 6724

Description: A function maps to its range iff the range is a subset of its codomain. Generalization of ffrn 6722. (Contributed by AV, 20-Sep-2024.)

Hypothesis

Ref	Expression
ffrnbd.r	⊢ (𝜑 → ran 𝐹 ⊆ 𝐵)

Assertion

Ref	Expression
ffrnbd	⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐴⟶ran 𝐹))

Proof of Theorem ffrnbd

Step	Hyp	Ref	Expression
1		ffrnb 6723	. 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹:𝐴⟶ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵))
2		ffrnbd.r	. . 3 ⊢ (𝜑 → ran 𝐹 ⊆ 𝐵)
3	2	biantrud 531	. 2 ⊢ (𝜑 → (𝐹:𝐴⟶ran 𝐹 ↔ (𝐹:𝐴⟶ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵)))
4	1, 3	bitr4id 290	1 ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐴⟶ran 𝐹))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ⊆ wss 3941  ran crn 5668  ⟶wf 6530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-in 3948  df-ss 3958  df-f 6538

This theorem is referenced by: (None)