Theorem rnmpt0f 6230

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 416	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝑉))
4	1, 3	ralrimi 3260	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉)
5		dmmptg 6229	. . . . 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 2768	. . 3 ⊢ (𝜑 → 𝐴 = dom (𝑥 ∈ 𝐴 ↦ 𝐵))
8	7	eqeq1d 2764	. 2 ⊢ (𝜑 → (𝐴 = ∅ ↔ dom (𝑥 ∈ 𝐴 ↦ 𝐵) = ∅))
9		dm0rn0 5900	. . 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 5913	. . . . 5 ⊢ ran 𝐹 = ran (𝑥 ∈ 𝐴 ↦ 𝐵)
13	12	a1i 11	. . . 4 ⊢ (𝜑 → ran 𝐹 = ran (𝑥 ∈ 𝐴 ↦ 𝐵))
14	13	eqcomd 2768	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) = ran 𝐹)
15	14	eqeq1d 2764	. 2 ⊢ (𝜑 → (ran (𝑥 ∈ 𝐴 ↦ 𝐵) = ∅ ↔ ran 𝐹 = ∅))
16	8, 10, 15	3bitrrd 308	1 ⊢ (𝜑 → (ran 𝐹 = ∅ ↔ 𝐴 = ∅))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1560  Ⅎwnf 1803   ∈ wcel 2142  ∀wral 3076  ∅c0 4285   ↦ cmpt 5181  dom cdm 5647  ran crn 5648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3077  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-mpt 5182  df-xp 5653  df-rel 5654  df-cnv 5655  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660

This theorem is referenced by:  rnmptn0  6231