Theorem ffrnbd 6616

Description: A function maps to its range iff the the range is a subset of its codomain. Generalization of ffrn 6614. (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 6615	. 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹:𝐴⟶ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵))
2		ffrnbd.r	. . 3 ⊢ (𝜑 → ran 𝐹 ⊆ 𝐵)
3	2	biantrud 532	. 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 396   ⊆ wss 3887  ran crn 5590  ⟶wf 6429

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-in 3894  df-ss 3904  df-f 6437

This theorem is referenced by: (None)