Theorem rnmptn0 40220

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
rnmptn0.x	⊢ Ⅎ𝑥𝜑
rnmptn0.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
rnmptn0.f	⊢ 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 3005	. . 3 ⊢ (𝜑 → ¬ 𝐴 = ∅)
3		rnmptn0.x	. . . 4 ⊢ Ⅎ𝑥𝜑
4		rnmptn0.b	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
5		rnmptn0.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
6	3, 4, 5	rnmpt0 40219	. . 3 ⊢ (𝜑 → (ran 𝐹 = ∅ ↔ 𝐴 = ∅))
7	2, 6	mtbird 317	. 2 ⊢ (𝜑 → ¬ ran 𝐹 = ∅)
8	7	neqned 3007	1 ⊢ (𝜑 → ran 𝐹 ≠ ∅)

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1658  Ⅎwnf 1884   ∈ wcel 2166   ≠ wne 3000  ∅c0 4145   ↦ cmpt 4953  ran crn 5344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pr 5128

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rab 3127  df-v 3417  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4875  df-opab 4937  df-mpt 4954  df-xp 5349  df-rel 5350  df-cnv 5351  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356

This theorem is referenced by:  infnsuprnmpt  40266  suprclrnmpt  40267  fisupclrnmpt  40418  supxrrernmpt  40444  suprleubrnmpt  40445  supxrre3rnmpt  40452  supminfrnmpt  40468  infrpgernmpt  40490  limsupvaluz2  40766  ioorrnopnlem  41316  iunhoiioolem  41684  vonioolem1  41689