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Theorem supp0 7583
Description: The support of the empty set is the empty set. (Contributed by AV, 12-Apr-2019.)
Assertion
Ref Expression
supp0 (𝑍𝑊 → (∅ supp 𝑍) = ∅)

Proof of Theorem supp0
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 0ex 5028 . . 3 ∅ ∈ V
2 suppval 7580 . . 3 ((∅ ∈ V ∧ 𝑍𝑊) → (∅ supp 𝑍) = {𝑖 ∈ dom ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}})
31, 2mpan 680 . 2 (𝑍𝑊 → (∅ supp 𝑍) = {𝑖 ∈ dom ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}})
4 dm0 5586 . . 3 dom ∅ = ∅
5 rabeq 3389 . . 3 (dom ∅ = ∅ → {𝑖 ∈ dom ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}} = {𝑖 ∈ ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}})
64, 5mp1i 13 . 2 (𝑍𝑊 → {𝑖 ∈ dom ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}} = {𝑖 ∈ ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}})
7 rab0 4186 . . 3 {𝑖 ∈ ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}} = ∅
87a1i 11 . 2 (𝑍𝑊 → {𝑖 ∈ ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}} = ∅)
93, 6, 83eqtrd 2818 1 (𝑍𝑊 → (∅ supp 𝑍) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  wcel 2107  wne 2969  {crab 3094  Vcvv 3398  c0 4141  {csn 4398  dom cdm 5357  cima 5360  (class class class)co 6924   supp csupp 7578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pr 5140  ax-un 7228
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-opab 4951  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-supp 7579
This theorem is referenced by:  0fsupp  8587  gsumval3  18698
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