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Theorem rnmptn0 38408
 Description: The range of a function in map-to notation is nonempty if the domain is nonempty. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
Hypotheses
Ref Expression
rnmptn0.x 𝑥𝜑
rnmptn0.b ((𝜑𝑥𝐴) → 𝐵𝑉)
rnmptn0.f 𝐹 = (𝑥𝐴𝐵)
rnmptn0.a (𝜑𝐴 ≠ ∅)
Assertion
Ref Expression
rnmptn0 (𝜑 → ran 𝐹 ≠ ∅)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem rnmptn0
StepHypRef Expression
1 rnmptn0.a . . . 4 (𝜑𝐴 ≠ ∅)
21neneqd 2787 . . 3 (𝜑 → ¬ 𝐴 = ∅)
3 rnmptn0.x . . . 4 𝑥𝜑
4 rnmptn0.b . . . 4 ((𝜑𝑥𝐴) → 𝐵𝑉)
5 rnmptn0.f . . . 4 𝐹 = (𝑥𝐴𝐵)
63, 4, 5rnmpt0 38407 . . 3 (𝜑 → (ran 𝐹 = ∅ ↔ 𝐴 = ∅))
72, 6mtbird 314 . 2 (𝜑 → ¬ ran 𝐹 = ∅)
87neqned 2789 1 (𝜑 → ran 𝐹 ≠ ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977   ≠ wne 2780  ∅c0 3874   ↦ cmpt 4643  ran crn 5039 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-mpt 4645  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051 This theorem is referenced by:  ioorrnopnlem  39200  iunhoiioolem  39566  vonioolem1  39571
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