Theorem supp0 8121

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 5257	. . 3 ⊢ ∅ ∈ V
2		suppval 8118	. . 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 5874	. . 3 ⊢ dom ∅ = ∅
5		rabeq 3417	. . 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 4345	. . 3 ⊢ {𝑖 ∈ ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}} = ∅
8	7	a1i 11	. 2 ⊢ (𝑍 ∈ 𝑊 → {𝑖 ∈ ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}} = ∅)
9	3, 6, 8	3eqtrd 2768	1 ⊢ (𝑍 ∈ 𝑊 → (∅ supp 𝑍) = ∅)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  {crab 3402  Vcvv 3444  ∅c0 4292  {csn 4585  dom cdm 5631   “ cima 5634  (class class class)co 7369   supp csupp 8116

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-supp 8117

This theorem is referenced by:  0fsupp  9317  gsumval3  19813