Theorem rnmptn0 6264

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 2964	. . 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 6263	. . 3 ⊢ (𝜑 → (ran 𝐹 = ∅ ↔ 𝐴 = ∅))
7	2, 6	mtbird 328	. 2 ⊢ (𝜑 → ¬ ran 𝐹 = ∅)
8	7	neqned 2966	1 ⊢ (𝜑 → ran 𝐹 ≠ ∅)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  Ⅎwnf 1816   ∈ wcel 2147   ≠ wne 2959  ∅c0 4281   ↦ cmpt 5192  ran crn 5659

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-mpt 5193  df-xp 5664  df-rel 5665  df-cnv 5666  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671

This theorem is used by:  nsgqusf1olem1  34068  suprnmpt  46462  infnsuprnmpt  46535  suprclrnmpt  46536  fisupclrnmpt  46682  supxrrernmpt  46704  suprleubrnmpt  46705  supxrre3rnmpt  46712  supminfrnmpt  46728  infrpgernmpt  46748  limsupvaluz2  47021  ioorrnopnlem  47587  iunhoiioolem  47958  vonioolem1  47963