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Theorem rnmptn0 41634
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 3011 . . 3 (𝜑 → ¬ 𝐴 = ∅)
3 rnmptn0.x . . . 4 𝑥𝜑
4 rnmptn0.b . . . 4 ((𝜑𝑥𝐴) → 𝐵𝑉)
5 rnmptn0.f . . . 4 𝐹 = (𝑥𝐴𝐵)
63, 4, 5rnmpt0 41633 . . 3 (𝜑 → (ran 𝐹 = ∅ ↔ 𝐴 = ∅))
72, 6mtbird 327 . 2 (𝜑 → ¬ ran 𝐹 = ∅)
87neqned 3013 1 (𝜑 → ran 𝐹 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1537  wnf 1784  wcel 2114  wne 3006  c0 4265  cmpt 5118  ran crn 5528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5175  ax-nul 5182  ax-pr 5302
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rab 3134  df-v 3472  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5039  df-opab 5101  df-mpt 5119  df-xp 5533  df-rel 5534  df-cnv 5535  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540
This theorem is referenced by:  infnsuprnmpt  41672  suprclrnmpt  41673  fisupclrnmpt  41821  supxrrernmpt  41845  suprleubrnmpt  41846  supxrre3rnmpt  41853  supminfrnmpt  41869  infrpgernmpt  41891  limsupvaluz2  42167  ioorrnopnlem  42733  iunhoiioolem  43101  vonioolem1  43106
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