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Theorem ffrnbd 6752
Description: A function maps to its range iff the range is a subset of its codomain. Generalization of ffrn 6750. (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 6751 . 2 (𝐹:𝐴𝐵 ↔ (𝐹:𝐴⟶ran 𝐹 ∧ ran 𝐹𝐵))
2 ffrnbd.r . . 3 (𝜑 → ran 𝐹𝐵)
32biantrud 531 . 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 206  wa 395  wss 3963  ran crn 5690  wf 6559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792
This theorem depends on definitions:  df-bi 207  df-an 396  df-ss 3980  df-f 6567
This theorem is referenced by: (None)
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