Theorem ffrnbd 6600

Description: A function maps to its range iff the the range is a subset of its codomain. Generalization of ffrn 6598. (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 6599	. 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 289	1 ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐴⟶ran 𝐹))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ⊆ wss 3883  ran crn 5581  ⟶wf 6414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-in 3890  df-ss 3900  df-f 6422

This theorem is referenced by: (None)