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Theorem rnmptn0 38884
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 2795 . . 3 (𝜑 → ¬ 𝐴 = ∅)
3 rnmptn0.x . . . 4 𝑥𝜑
4 rnmptn0.b . . . 4 ((𝜑𝑥𝐴) → 𝐵𝑉)
5 rnmptn0.f . . . 4 𝐹 = (𝑥𝐴𝐵)
63, 4, 5rnmpt0 38883 . . 3 (𝜑 → (ran 𝐹 = ∅ ↔ 𝐴 = ∅))
72, 6mtbird 315 . 2 (𝜑 → ¬ ran 𝐹 = ∅)
87neqned 2797 1 (𝜑 → ran 𝐹 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wnf 1705  wcel 1987  wne 2790  c0 3891  cmpt 4673  ran crn 5075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-mpt 4675  df-xp 5080  df-rel 5081  df-cnv 5082  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087
This theorem is referenced by:  infnsuprnmpt  38938  suprclrnmpt  38939  fisupclrnmpt  39083  supxrrernmpt  39109  suprleubrnmpt  39110  supxrre3rnmpt  39117  limsupvaluz2  39371  ioorrnopnlem  39828  iunhoiioolem  40193  vonioolem1  40198
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