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Theorem supp0 8189
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 5313 . . 3 ∅ ∈ V
2 suppval 8186 . . 3 ((∅ ∈ V ∧ 𝑍𝑊) → (∅ supp 𝑍) = {𝑖 ∈ dom ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}})
31, 2mpan 690 . 2 (𝑍𝑊 → (∅ supp 𝑍) = {𝑖 ∈ dom ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}})
4 dm0 5934 . . 3 dom ∅ = ∅
5 rabeq 3448 . . 3 (dom ∅ = ∅ → {𝑖 ∈ dom ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}} = {𝑖 ∈ ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}})
64, 5mp1i 13 . 2 (𝑍𝑊 → {𝑖 ∈ dom ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}} = {𝑖 ∈ ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}})
7 rab0 4392 . . 3 {𝑖 ∈ ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}} = ∅
87a1i 11 . 2 (𝑍𝑊 → {𝑖 ∈ ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}} = ∅)
93, 6, 83eqtrd 2779 1 (𝑍𝑊 → (∅ supp 𝑍) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  wne 2938  {crab 3433  Vcvv 3478  c0 4339  {csn 4631  dom cdm 5689  cima 5692  (class class class)co 7431   supp csupp 8184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-supp 8185
This theorem is referenced by:  0fsupp  9428  gsumval3  19940
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