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Theorem ffrnb 6529
Description: Characterization of a function with domain and codomain (essentially using that the range is always included in the codomain). Generalization of ffrn 6528. (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 6353 . 2 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
2 dffn3 6527 . . 3 (𝐹 Fn 𝐴𝐹:𝐴⟶ran 𝐹)
32anbi1i 627 . 2 ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) ↔ (𝐹:𝐴⟶ran 𝐹 ∧ ran 𝐹𝐵))
41, 3bitri 278 1 (𝐹:𝐴𝐵 ↔ (𝐹:𝐴⟶ran 𝐹 ∧ ran 𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wss 3853  ran crn 5536   Fn wfn 6344  wf 6345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2711
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-v 3402  df-in 3860  df-ss 3870  df-f 6353
This theorem is referenced by:  ffrnbd  6530
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