Theorem rnmptn0 6245

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 2963	. . 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 6244	. . 3 ⊢ (𝜑 → (ran 𝐹 = ∅ ↔ 𝐴 = ∅))
7	2, 6	mtbird 328	. 2 ⊢ (𝜑 → ¬ ran 𝐹 = ∅)
8	7	neqned 2965	1 ⊢ (𝜑 → ran 𝐹 ≠ ∅)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570  Ⅎwnf 1813   ∈ wcel 2143   ≠ wne 2958  ∅c0 4286   ↦ cmpt 5192  ran crn 5662

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-mpt 5193  df-xp 5667  df-rel 5668  df-cnv 5669  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674

This theorem is referenced by:  nsgqusf1olem1  33722  suprnmpt  45892  infnsuprnmpt  45965  suprclrnmpt  45966  fisupclrnmpt  46113  supxrrernmpt  46135  suprleubrnmpt  46136  supxrre3rnmpt  46143  supminfrnmpt  46159  infrpgernmpt  46179  limsupvaluz2  46452  ioorrnopnlem  47018  iunhoiioolem  47389  vonioolem1  47394