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Theorem rnmptn0 6200
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 2945 . . 3 (𝜑 → ¬ 𝐴 = ∅)
3 rnmpt0f.1 . . . 4 𝑥𝜑
4 rnmpt0f.2 . . . 4 ((𝜑𝑥𝐴) → 𝐵𝑉)
5 rnmpt0f.3 . . . 4 𝐹 = (𝑥𝐴𝐵)
63, 4, 5rnmpt0f 6199 . . 3 (𝜑 → (ran 𝐹 = ∅ ↔ 𝐴 = ∅))
72, 6mtbird 325 . 2 (𝜑 → ¬ ran 𝐹 = ∅)
87neqned 2947 1 (𝜑 → ran 𝐹 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wnf 1786  wcel 2107  wne 2940  c0 4286  cmpt 5192  ran crn 5638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-mpt 5193  df-xp 5643  df-rel 5644  df-cnv 5645  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650
This theorem is referenced by:  nsgqusf1olem1  32246  suprnmpt  43483  infnsuprnmpt  43569  suprclrnmpt  43570  fisupclrnmpt  43723  supxrrernmpt  43746  suprleubrnmpt  43747  supxrre3rnmpt  43754  supminfrnmpt  43770  infrpgernmpt  43790  limsupvaluz2  44069  ioorrnopnlem  44635  iunhoiioolem  45006  vonioolem1  45011
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