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Theorem rnmptn0 6233
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 2937 . . 3 (𝜑 → ¬ 𝐴 = ∅)
3 rnmpt0f.1 . . . 4 𝑥𝜑
4 rnmpt0f.2 . . . 4 ((𝜑𝑥𝐴) → 𝐵𝑉)
5 rnmpt0f.3 . . . 4 𝐹 = (𝑥𝐴𝐵)
63, 4, 5rnmpt0f 6232 . . 3 (𝜑 → (ran 𝐹 = ∅ ↔ 𝐴 = ∅))
72, 6mtbird 325 . 2 (𝜑 → ¬ ran 𝐹 = ∅)
87neqned 2939 1 (𝜑 → ran 𝐹 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wnf 1777  wcel 2098  wne 2932  c0 4314  cmpt 5221  ran crn 5667
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 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-mpt 5222  df-xp 5672  df-rel 5673  df-cnv 5674  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679
This theorem is referenced by:  nsgqusf1olem1  32993  suprnmpt  44358  infnsuprnmpt  44439  suprclrnmpt  44440  fisupclrnmpt  44593  supxrrernmpt  44616  suprleubrnmpt  44617  supxrre3rnmpt  44624  supminfrnmpt  44640  infrpgernmpt  44660  limsupvaluz2  44939  ioorrnopnlem  45505  iunhoiioolem  45876  vonioolem1  45881
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