Theorem rnmpt0f 6242

Description: The range of a function in maps-to notation is empty if and only if its domain is empty. (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)

Assertion

Ref	Expression
rnmpt0f	⊢ (𝜑 → (ran 𝐹 = ∅ ↔ 𝐴 = ∅))

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem rnmpt0f

Step	Hyp	Ref	Expression
1		rnmpt0f.1	. . . . . 6 ⊢ Ⅎ𝑥𝜑
2		rnmpt0f.2	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
3	2	ex 413	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝑉))
4	1, 3	ralrimi 3254	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉)
5		dmmptg 6241	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴)
6	4, 5	syl 17	. . . 4 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴)
7	6	eqcomd 2738	. . 3 ⊢ (𝜑 → 𝐴 = dom (𝑥 ∈ 𝐴 ↦ 𝐵))
8	7	eqeq1d 2734	. 2 ⊢ (𝜑 → (𝐴 = ∅ ↔ dom (𝑥 ∈ 𝐴 ↦ 𝐵) = ∅))
9		dm0rn0 5924	. . 3 ⊢ (dom (𝑥 ∈ 𝐴 ↦ 𝐵) = ∅ ↔ ran (𝑥 ∈ 𝐴 ↦ 𝐵) = ∅)
10	9	a1i 11	. 2 ⊢ (𝜑 → (dom (𝑥 ∈ 𝐴 ↦ 𝐵) = ∅ ↔ ran (𝑥 ∈ 𝐴 ↦ 𝐵) = ∅))
11		rnmpt0f.3	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
12	11	rneqi 5936	. . . . 5 ⊢ ran 𝐹 = ran (𝑥 ∈ 𝐴 ↦ 𝐵)
13	12	a1i 11	. . . 4 ⊢ (𝜑 → ran 𝐹 = ran (𝑥 ∈ 𝐴 ↦ 𝐵))
14	13	eqcomd 2738	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) = ran 𝐹)
15	14	eqeq1d 2734	. 2 ⊢ (𝜑 → (ran (𝑥 ∈ 𝐴 ↦ 𝐵) = ∅ ↔ ran 𝐹 = ∅))
16	8, 10, 15	3bitrrd 305	1 ⊢ (𝜑 → (ran 𝐹 = ∅ ↔ 𝐴 = ∅))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541  Ⅎwnf 1785   ∈ wcel 2106  ∀wral 3061  ∅c0 4322   ↦ cmpt 5231  dom cdm 5676  ran crn 5677

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-mpt 5232  df-xp 5682  df-rel 5683  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689

This theorem is referenced by:  rnmptn0  6243