Theorem supp0 8190

Description: The support of the empty set is the empty set. (Contributed by AV, 12-Apr-2019.)

Assertion

Ref	Expression
supp0	⊢ (𝑍 ∈ 𝑊 → (∅ supp 𝑍) = ∅)

Proof of Theorem supp0

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ex 5307	. . 3 ⊢ ∅ ∈ V
2		suppval 8187	. . 3 ⊢ ((∅ ∈ V ∧ 𝑍 ∈ 𝑊) → (∅ supp 𝑍) = {𝑖 ∈ dom ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}})
3	1, 2	mpan 690	. 2 ⊢ (𝑍 ∈ 𝑊 → (∅ supp 𝑍) = {𝑖 ∈ dom ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}})
4		dm0 5931	. . 3 ⊢ dom ∅ = ∅
5		rabeq 3451	. . 3 ⊢ (dom ∅ = ∅ → {𝑖 ∈ dom ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}} = {𝑖 ∈ ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}})
6	4, 5	mp1i 13	. 2 ⊢ (𝑍 ∈ 𝑊 → {𝑖 ∈ dom ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}} = {𝑖 ∈ ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}})
7		rab0 4386	. . 3 ⊢ {𝑖 ∈ ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}} = ∅
8	7	a1i 11	. 2 ⊢ (𝑍 ∈ 𝑊 → {𝑖 ∈ ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}} = ∅)
9	3, 6, 8	3eqtrd 2781	1 ⊢ (𝑍 ∈ 𝑊 → (∅ supp 𝑍) = ∅)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  {crab 3436  Vcvv 3480  ∅c0 4333  {csn 4626  dom cdm 5685   “ cima 5688  (class class class)co 7431   supp csupp 8185

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-supp 8186

This theorem is referenced by:  0fsupp  9430  gsumval3  19925