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Theorem ffrnb 6615
Description: Characterization of a function with domain and codomain (essentially using that the range is always included in the codomain). Generalization of ffrn 6614. (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 6437 . 2 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
2 dffn3 6613 . . 3 (𝐹 Fn 𝐴𝐹:𝐴⟶ran 𝐹)
32anbi1i 624 . 2 ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) ↔ (𝐹:𝐴⟶ran 𝐹 ∧ ran 𝐹𝐵))
41, 3bitri 274 1 (𝐹:𝐴𝐵 ↔ (𝐹:𝐴⟶ran 𝐹 ∧ ran 𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396  wss 3887  ran crn 5590   Fn wfn 6428  wf 6429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-in 3894  df-ss 3904  df-f 6437
This theorem is referenced by:  ffrnbd  6616
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