Theorem ffrnb 6549

Description: Characterization of a function with domain and codomain (essentially using that the range is always included in the codomain). Generalization of ffrn 6548. (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 6373	. 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
2		dffn3 6547	. . 3 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)
3	2	anbi1i 627	. 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 399   ⊆ wss 3857  ran crn 5541   Fn wfn 6364  ⟶wf 6365

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2706

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2713  df-cleq 2726  df-clel 2812  df-v 3403  df-in 3864  df-ss 3874  df-f 6373

This theorem is referenced by:  ffrnbd  6550