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Theorem ffrnbd 6616
Description: A function maps to its range iff the the range is a subset of its codomain. Generalization of ffrn 6614. (Contributed by AV, 20-Sep-2024.)
Hypothesis
Ref Expression
ffrnbd.r (𝜑 → ran 𝐹𝐵)
Assertion
Ref Expression
ffrnbd (𝜑 → (𝐹:𝐴𝐵𝐹:𝐴⟶ran 𝐹))

Proof of Theorem ffrnbd
StepHypRef Expression
1 ffrnb 6615 . 2 (𝐹:𝐴𝐵 ↔ (𝐹:𝐴⟶ran 𝐹 ∧ ran 𝐹𝐵))
2 ffrnbd.r . . 3 (𝜑 → ran 𝐹𝐵)
32biantrud 532 . 2 (𝜑 → (𝐹:𝐴⟶ran 𝐹 ↔ (𝐹:𝐴⟶ran 𝐹 ∧ ran 𝐹𝐵)))
41, 3bitr4id 290 1 (𝜑 → (𝐹:𝐴𝐵𝐹:𝐴⟶ran 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wss 3887  ran crn 5590  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: (None)
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