Theorem ffrnb 6732

Description: Characterization of a function with domain and codomain (essentially using that the range is always included in the codomain). Generalization of ffrn 6731. (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 6547	. 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
2		dffn3 6730	. . 3 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)
3	2	anbi1i 624	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) ↔ (𝐹:𝐴⟶ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵))
4	1, 3	bitri 274	1 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹:𝐴⟶ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵))

Syntax hints:   ↔ wb 205   ∧ wa 396   ⊆ wss 3948  ran crn 5677   Fn wfn 6538  ⟶wf 6539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-in 3955  df-ss 3965  df-f 6547

This theorem is referenced by:  ffrnbd  6733