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Theorem supp0 7456
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 4930 . . 3 ∅ ∈ V
2 suppval 7453 . . 3 ((∅ ∈ V ∧ 𝑍𝑊) → (∅ supp 𝑍) = {𝑖 ∈ dom ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}})
31, 2mpan 708 . 2 (𝑍𝑊 → (∅ supp 𝑍) = {𝑖 ∈ dom ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}})
4 dm0 5482 . . 3 dom ∅ = ∅
5 rabeq 3320 . . 3 (dom ∅ = ∅ → {𝑖 ∈ dom ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}} = {𝑖 ∈ ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}})
64, 5mp1i 13 . 2 (𝑍𝑊 → {𝑖 ∈ dom ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}} = {𝑖 ∈ ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}})
7 rab0 4086 . . 3 {𝑖 ∈ ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}} = ∅
87a1i 11 . 2 (𝑍𝑊 → {𝑖 ∈ ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}} = ∅)
93, 6, 83eqtrd 2786 1 (𝑍𝑊 → (∅ supp 𝑍) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1620  wcel 2127  wne 2920  {crab 3042  Vcvv 3328  c0 4046  {csn 4309  dom cdm 5254  cima 5257  (class class class)co 6801   supp csupp 7451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-sbc 3565  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-br 4793  df-opab 4853  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-iota 6000  df-fun 6039  df-fv 6045  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-supp 7452
This theorem is referenced by:  0fsupp  8450  gsumval3  18479
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