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Theorem ffrnbd 6530
Description: A function maps to its range iff the the range is a subset of its codomain. Generalization of ffrn 6528. (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 6529 . 2 (𝐹:𝐴𝐵 ↔ (𝐹:𝐴⟶ran 𝐹 ∧ ran 𝐹𝐵))
2 ffrnbd.r . . 3 (𝜑 → ran 𝐹𝐵)
32biantrud 535 . 2 (𝜑 → (𝐹:𝐴⟶ran 𝐹 ↔ (𝐹:𝐴⟶ran 𝐹 ∧ ran 𝐹𝐵)))
41, 3bitr4id 293 1 (𝜑 → (𝐹:𝐴𝐵𝐹:𝐴⟶ran 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wss 3853  ran crn 5536  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: (None)
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