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Theorem ffrnb 6684
Description: Characterization of a function with domain and codomain (essentially using that the range is always included in the codomain). Generalization of ffrn 6683. (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 6501 . 2 (𝐹:𝐴⟢𝐡 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 βŠ† 𝐡))
2 dffn3 6682 . . 3 (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟢ran 𝐹)
32anbi1i 625 . 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 397   βŠ† wss 3911  ran crn 5635   Fn wfn 6492  βŸΆwf 6493
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3446  df-in 3918  df-ss 3928  df-f 6501
This theorem is referenced by:  ffrnbd  6685
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