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Theorem supp0 8170
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 5307 . . 3 ∅ ∈ V
2 suppval 8167 . . 3 ((∅ ∈ V ∧ 𝑍𝑊) → (∅ supp 𝑍) = {𝑖 ∈ dom ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}})
31, 2mpan 689 . 2 (𝑍𝑊 → (∅ supp 𝑍) = {𝑖 ∈ dom ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}})
4 dm0 5923 . . 3 dom ∅ = ∅
5 rabeq 3443 . . 3 (dom ∅ = ∅ → {𝑖 ∈ dom ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}} = {𝑖 ∈ ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}})
64, 5mp1i 13 . 2 (𝑍𝑊 → {𝑖 ∈ dom ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}} = {𝑖 ∈ ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}})
7 rab0 4383 . . 3 {𝑖 ∈ ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}} = ∅
87a1i 11 . 2 (𝑍𝑊 → {𝑖 ∈ ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}} = ∅)
93, 6, 83eqtrd 2772 1 (𝑍𝑊 → (∅ supp 𝑍) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  wne 2937  {crab 3429  Vcvv 3471  c0 4323  {csn 4629  dom cdm 5678  cima 5681  (class class class)co 7420   supp csupp 8165
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-supp 8166
This theorem is referenced by:  0fsupp  9413  gsumval3  19861
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