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Theorem suppval1 7835
Description: The value of the operation constructing the support of a function. (Contributed by AV, 6-Apr-2019.)
Assertion
Ref Expression
suppval1 ((Fun 𝑋𝑋𝑉𝑍𝑊) → (𝑋 supp 𝑍) = {𝑖 ∈ dom 𝑋 ∣ (𝑋𝑖) ≠ 𝑍})
Distinct variable groups:   𝑖,𝑉   𝑖,𝑊   𝑖,𝑋   𝑖,𝑍

Proof of Theorem suppval1
StepHypRef Expression
1 suppval 7831 . . 3 ((𝑋𝑉𝑍𝑊) → (𝑋 supp 𝑍) = {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}})
213adant1 1126 . 2 ((Fun 𝑋𝑋𝑉𝑍𝑊) → (𝑋 supp 𝑍) = {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}})
3 funfn 6384 . . . . . . . . 9 (Fun 𝑋𝑋 Fn dom 𝑋)
43biimpi 218 . . . . . . . 8 (Fun 𝑋𝑋 Fn dom 𝑋)
543ad2ant1 1129 . . . . . . 7 ((Fun 𝑋𝑋𝑉𝑍𝑊) → 𝑋 Fn dom 𝑋)
6 fnsnfv 6742 . . . . . . 7 ((𝑋 Fn dom 𝑋𝑖 ∈ dom 𝑋) → {(𝑋𝑖)} = (𝑋 “ {𝑖}))
75, 6sylan 582 . . . . . 6 (((Fun 𝑋𝑋𝑉𝑍𝑊) ∧ 𝑖 ∈ dom 𝑋) → {(𝑋𝑖)} = (𝑋 “ {𝑖}))
87eqcomd 2827 . . . . 5 (((Fun 𝑋𝑋𝑉𝑍𝑊) ∧ 𝑖 ∈ dom 𝑋) → (𝑋 “ {𝑖}) = {(𝑋𝑖)})
98neeq1d 3075 . . . 4 (((Fun 𝑋𝑋𝑉𝑍𝑊) ∧ 𝑖 ∈ dom 𝑋) → ((𝑋 “ {𝑖}) ≠ {𝑍} ↔ {(𝑋𝑖)} ≠ {𝑍}))
10 fvex 6682 . . . . . 6 (𝑋𝑖) ∈ V
11 sneqbg 4773 . . . . . 6 ((𝑋𝑖) ∈ V → ({(𝑋𝑖)} = {𝑍} ↔ (𝑋𝑖) = 𝑍))
1210, 11mp1i 13 . . . . 5 (((Fun 𝑋𝑋𝑉𝑍𝑊) ∧ 𝑖 ∈ dom 𝑋) → ({(𝑋𝑖)} = {𝑍} ↔ (𝑋𝑖) = 𝑍))
1312necon3bid 3060 . . . 4 (((Fun 𝑋𝑋𝑉𝑍𝑊) ∧ 𝑖 ∈ dom 𝑋) → ({(𝑋𝑖)} ≠ {𝑍} ↔ (𝑋𝑖) ≠ 𝑍))
149, 13bitrd 281 . . 3 (((Fun 𝑋𝑋𝑉𝑍𝑊) ∧ 𝑖 ∈ dom 𝑋) → ((𝑋 “ {𝑖}) ≠ {𝑍} ↔ (𝑋𝑖) ≠ 𝑍))
1514rabbidva 3478 . 2 ((Fun 𝑋𝑋𝑉𝑍𝑊) → {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}} = {𝑖 ∈ dom 𝑋 ∣ (𝑋𝑖) ≠ 𝑍})
162, 15eqtrd 2856 1 ((Fun 𝑋𝑋𝑉𝑍𝑊) → (𝑋 supp 𝑍) = {𝑖 ∈ dom 𝑋 ∣ (𝑋𝑖) ≠ 𝑍})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1083   = wceq 1533  wcel 2110  wne 3016  {crab 3142  Vcvv 3494  {csn 4566  dom cdm 5554  cima 5557  Fun wfun 6348   Fn wfn 6349  cfv 6354  (class class class)co 7155   supp csupp 7829
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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329  ax-un 7460
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-fv 6362  df-ov 7158  df-oprab 7159  df-mpo 7160  df-supp 7830
This theorem is referenced by:  suppvalfn  7836  suppfnss  7854  fnsuppres  7856  rmfsupp2  30866  domnmsuppn0  44416  rmsuppss  44417  mndpsuppss  44418  scmsuppss  44419  suppdm  44564
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