Theorem rnmptn0 6147

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	⊢ Ⅎ𝑥𝜑
rnmpt0f.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
rnmpt0f.3	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
rnmptn0.a	⊢ (𝜑 → 𝐴 ≠ ∅)

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

Step	Hyp	Ref	Expression
1		rnmptn0.a	. . . 4 ⊢ (𝜑 → 𝐴 ≠ ∅)
2	1	neneqd 2948	. . 3 ⊢ (𝜑 → ¬ 𝐴 = ∅)
3		rnmpt0f.1	. . . 4 ⊢ Ⅎ𝑥𝜑
4		rnmpt0f.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
5		rnmpt0f.3	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
6	3, 4, 5	rnmpt0f 6146	. . 3 ⊢ (𝜑 → (ran 𝐹 = ∅ ↔ 𝐴 = ∅))
7	2, 6	mtbird 325	. 2 ⊢ (𝜑 → ¬ ran 𝐹 = ∅)
8	7	neqned 2950	1 ⊢ (𝜑 → ran 𝐹 ≠ ∅)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539  Ⅎwnf 1786   ∈ wcel 2106   ≠ wne 2943  ∅c0 4256   ↦ cmpt 5157  ran crn 5590

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-mpt 5158  df-xp 5595  df-rel 5596  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602

This theorem is referenced by:  nsgqusf1olem1  31598  suprnmpt  42710  infnsuprnmpt  42796  suprclrnmpt  42797  fisupclrnmpt  42938  supxrrernmpt  42961  suprleubrnmpt  42962  supxrre3rnmpt  42969  supminfrnmpt  42985  infrpgernmpt  43005  limsupvaluz2  43279  ioorrnopnlem  43845  iunhoiioolem  44213  vonioolem1  44218