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Theorem ffrnb 6723
Description: Characterization of a function with domain and codomain (essentially using that the range is always included in the codomain). Generalization of ffrn 6722. (Contributed by BJ, 21-Sep-2024.)
Assertion
Ref Expression
ffrnb (𝐹:𝐴⟢𝐡 ↔ (𝐹:𝐴⟢ran 𝐹 ∧ ran 𝐹 βŠ† 𝐡))

Proof of Theorem ffrnb
StepHypRef Expression
1 df-f 6538 . 2 (𝐹:𝐴⟢𝐡 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 βŠ† 𝐡))
2 dffn3 6721 . . 3 (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟢ran 𝐹)
32anbi1i 623 . 2 ((𝐹 Fn 𝐴 ∧ ran 𝐹 βŠ† 𝐡) ↔ (𝐹:𝐴⟢ran 𝐹 ∧ ran 𝐹 βŠ† 𝐡))
41, 3bitri 275 1 (𝐹:𝐴⟢𝐡 ↔ (𝐹:𝐴⟢ran 𝐹 ∧ ran 𝐹 βŠ† 𝐡))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 395   βŠ† wss 3941  ran crn 5668   Fn wfn 6529  βŸΆwf 6530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-in 3948  df-ss 3958  df-f 6538
This theorem is referenced by:  ffrnbd  6724
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