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Theorem rnmptn0 6251
Description: The range of a function in maps-to notation is nonempty if the domain is nonempty. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
Hypotheses
Ref Expression
rnmpt0f.1 𝑥𝜑
rnmpt0f.2 ((𝜑𝑥𝐴) → 𝐵𝑉)
rnmpt0f.3 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 2941 . . 3 (𝜑 → ¬ 𝐴 = ∅)
3 rnmpt0f.1 . . . 4 𝑥𝜑
4 rnmpt0f.2 . . . 4 ((𝜑𝑥𝐴) → 𝐵𝑉)
5 rnmpt0f.3 . . . 4 𝐹 = (𝑥𝐴𝐵)
63, 4, 5rnmpt0f 6250 . . 3 (𝜑 → (ran 𝐹 = ∅ ↔ 𝐴 = ∅))
72, 6mtbird 324 . 2 (𝜑 → ¬ ran 𝐹 = ∅)
87neqned 2943 1 (𝜑 → ran 𝐹 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wnf 1777  wcel 2098  wne 2936  c0 4324  cmpt 5233  ran crn 5681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5151  df-opab 5213  df-mpt 5234  df-xp 5686  df-rel 5687  df-cnv 5688  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693
This theorem is referenced by:  nsgqusf1olem1  33141  suprnmpt  44550  infnsuprnmpt  44628  suprclrnmpt  44629  fisupclrnmpt  44782  supxrrernmpt  44805  suprleubrnmpt  44806  supxrre3rnmpt  44813  supminfrnmpt  44829  infrpgernmpt  44849  limsupvaluz2  45128  ioorrnopnlem  45694  iunhoiioolem  46065  vonioolem1  46070
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