Theorem ffrnb 6724

Description: Characterization of a function with domain and codomain (essentially using that the range is always included in the codomain). Generalization of ffrn 6723. (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 6544	. 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
2		dffn3 6722	. . 3 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)
3	2	anbi1i 636	. 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))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ⊆ wss 3906  ran crn 5664   Fn wfn 6535  ⟶wf 6536

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828

This proof depends on definitions:  df-bi 210  df-an 402  df-ss 3923  df-f 6544

This theorem is used by:  ffrnbd  6725