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Theorem rnmptn0 40220
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
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 3005 . . 3 (𝜑 → ¬ 𝐴 = ∅)
3 rnmptn0.x . . . 4 𝑥𝜑
4 rnmptn0.b . . . 4 ((𝜑𝑥𝐴) → 𝐵𝑉)
5 rnmptn0.f . . . 4 𝐹 = (𝑥𝐴𝐵)
63, 4, 5rnmpt0 40219 . . 3 (𝜑 → (ran 𝐹 = ∅ ↔ 𝐴 = ∅))
72, 6mtbird 317 . 2 (𝜑 → ¬ ran 𝐹 = ∅)
87neqned 3007 1 (𝜑 → ran 𝐹 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1658  wnf 1884  wcel 2166  wne 3000  c0 4145  cmpt 4953  ran crn 5344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pr 5128
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rab 3127  df-v 3417  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4875  df-opab 4937  df-mpt 4954  df-xp 5349  df-rel 5350  df-cnv 5351  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356
This theorem is referenced by:  infnsuprnmpt  40266  suprclrnmpt  40267  fisupclrnmpt  40418  supxrrernmpt  40444  suprleubrnmpt  40445  supxrre3rnmpt  40452  supminfrnmpt  40468  infrpgernmpt  40490  limsupvaluz2  40766  ioorrnopnlem  41316  iunhoiioolem  41684  vonioolem1  41689
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