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Theorem supp0 7297
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 4788 . . 3 ∅ ∈ V
2 suppval 7294 . . 3 ((∅ ∈ V ∧ 𝑍𝑊) → (∅ supp 𝑍) = {𝑖 ∈ dom ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}})
31, 2mpan 706 . 2 (𝑍𝑊 → (∅ supp 𝑍) = {𝑖 ∈ dom ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}})
4 dm0 5337 . . 3 dom ∅ = ∅
5 rabeq 3190 . . 3 (dom ∅ = ∅ → {𝑖 ∈ dom ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}} = {𝑖 ∈ ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}})
64, 5mp1i 13 . 2 (𝑍𝑊 → {𝑖 ∈ dom ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}} = {𝑖 ∈ ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}})
7 rab0 3953 . . 3 {𝑖 ∈ ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}} = ∅
87a1i 11 . 2 (𝑍𝑊 → {𝑖 ∈ ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}} = ∅)
93, 6, 83eqtrd 2659 1 (𝑍𝑊 → (∅ supp 𝑍) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1482  wcel 1989  wne 2793  {crab 2915  Vcvv 3198  c0 3913  {csn 4175  dom cdm 5112  cima 5115  (class class class)co 6647   supp csupp 7292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-supp 7293
This theorem is referenced by:  0fsupp  8294  gsumval3  18302
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