Theorem rnmptn0 6136

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 2947	. . 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 6135	. . 3 ⊢ (𝜑 → (ran 𝐹 = ∅ ↔ 𝐴 = ∅))
7	2, 6	mtbird 324	. 2 ⊢ (𝜑 → ¬ ran 𝐹 = ∅)
8	7	neqned 2949	1 ⊢ (𝜑 → ran 𝐹 ≠ ∅)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539  Ⅎwnf 1787   ∈ wcel 2108   ≠ wne 2942  ∅c0 4253   ↦ cmpt 5153  ran crn 5581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-mpt 5154  df-xp 5586  df-rel 5587  df-cnv 5588  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593

This theorem is referenced by:  nsgqusf1olem1  31500  suprnmpt  42599  infnsuprnmpt  42685  suprclrnmpt  42686  fisupclrnmpt  42828  supxrrernmpt  42851  suprleubrnmpt  42852  supxrre3rnmpt  42859  supminfrnmpt  42875  infrpgernmpt  42895  limsupvaluz2  43169  ioorrnopnlem  43735  iunhoiioolem  44103  vonioolem1  44108