Theorem ffrnb 6721

Description: Characterization of a function with domain and codomain (essentially using that the range is always included in the codomain). Generalization of ffrn 6720. (Contributed by BJ, 21-Sep-2024.)

Assertion

Ref	Expression
ffrnb	⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹:𝐴⟶ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵))

Proof of Theorem ffrnb

Step	Hyp	Ref	Expression
1		df-f 6541	. 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
2		dffn3 6719	. . 3 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)
3	2	anbi1i 635	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) ↔ (𝐹:𝐴⟶ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵))
4	1, 3	bitri 278	1 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹:𝐴⟶ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵))

Syntax hints:   ↔ wb 209   ∧ wa 400   ⊆ wss 3913  ran crn 5663   Fn wfn 6532  ⟶wf 6533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822

This theorem depends on definitions:  df-bi 210  df-an 401  df-ss 3930  df-f 6541

This theorem is referenced by:  ffrnbd  6722