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Theorem rnmptn0 39931
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 2948 . . 3 (𝜑 → ¬ 𝐴 = ∅)
3 rnmptn0.x . . . 4 𝑥𝜑
4 rnmptn0.b . . . 4 ((𝜑𝑥𝐴) → 𝐵𝑉)
5 rnmptn0.f . . . 4 𝐹 = (𝑥𝐴𝐵)
63, 4, 5rnmpt0 39930 . . 3 (𝜑 → (ran 𝐹 = ∅ ↔ 𝐴 = ∅))
72, 6mtbird 314 . 2 (𝜑 → ¬ ran 𝐹 = ∅)
87neqned 2950 1 (𝜑 → ran 𝐹 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wnf 1856  wcel 2145  wne 2943  c0 4063  cmpt 4863  ran crn 5250
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-mpt 4864  df-xp 5255  df-rel 5256  df-cnv 5257  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262
This theorem is referenced by:  infnsuprnmpt  39983  suprclrnmpt  39984  fisupclrnmpt  40138  supxrrernmpt  40164  suprleubrnmpt  40165  supxrre3rnmpt  40172  supminfrnmpt  40188  infrpgernmpt  40211  limsupvaluz2  40488  ioorrnopnlem  41041  iunhoiioolem  41409  vonioolem1  41414
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