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Theorem ffrnb 6751
Description: Characterization of a function with domain and codomain (essentially using that the range is always included in the codomain). Generalization of ffrn 6750. (Contributed by BJ, 21-Sep-2024.)
Assertion
Ref Expression
ffrnb (𝐹:𝐴𝐵 ↔ (𝐹:𝐴⟶ran 𝐹 ∧ ran 𝐹𝐵))

Proof of Theorem ffrnb
StepHypRef Expression
1 df-f 6567 . 2 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
2 dffn3 6749 . . 3 (𝐹 Fn 𝐴𝐹:𝐴⟶ran 𝐹)
32anbi1i 624 . 2 ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) ↔ (𝐹:𝐴⟶ran 𝐹 ∧ ran 𝐹𝐵))
41, 3bitri 275 1 (𝐹:𝐴𝐵 ↔ (𝐹:𝐴⟶ran 𝐹 ∧ ran 𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wss 3963  ran crn 5690   Fn wfn 6558  wf 6559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792
This theorem depends on definitions:  df-bi 207  df-an 396  df-ss 3980  df-f 6567
This theorem is referenced by:  ffrnbd  6752
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