Theorem ffrnbd 6707

Description: A function maps to its range iff the range is a subset of its codomain. Generalization of ffrn 6705. (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 6706	. 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹:𝐴⟶ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵))
2		ffrnbd.r	. . 3 ⊢ (𝜑 → ran 𝐹 ⊆ 𝐵)
3	2	biantrud 539	. 2 ⊢ (𝜑 → (𝐹:𝐴⟶ran 𝐹 ↔ (𝐹:𝐴⟶ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵)))
4	1, 3	bitr4id 292	1 ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐴⟶ran 𝐹))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ⊆ wss 3904  ran crn 5648  ⟶wf 6517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815

This theorem depends on definitions:  df-bi 209  df-an 400  df-ss 3921  df-f 6525

This theorem is referenced by: (None)