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Theorem suppvalfn 8193
Description: The value of the operation constructing the support of a function with a given domain. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by AV, 22-Apr-2019.)
Assertion
Ref Expression
suppvalfn ((𝐹 Fn 𝑋𝑋𝑉𝑍𝑊) → (𝐹 supp 𝑍) = {𝑖𝑋 ∣ (𝐹𝑖) ≠ 𝑍})
Distinct variable groups:   𝑖,𝑉   𝑖,𝑊   𝑖,𝑋   𝑖,𝑍   𝑖,𝐹

Proof of Theorem suppvalfn
StepHypRef Expression
1 fnfun 6668 . . . 4 (𝐹 Fn 𝑋 → Fun 𝐹)
213ad2ant1 1134 . . 3 ((𝐹 Fn 𝑋𝑋𝑉𝑍𝑊) → Fun 𝐹)
3 fnex 7237 . . . 4 ((𝐹 Fn 𝑋𝑋𝑉) → 𝐹 ∈ V)
433adant3 1133 . . 3 ((𝐹 Fn 𝑋𝑋𝑉𝑍𝑊) → 𝐹 ∈ V)
5 simp3 1139 . . 3 ((𝐹 Fn 𝑋𝑋𝑉𝑍𝑊) → 𝑍𝑊)
6 suppval1 8191 . . 3 ((Fun 𝐹𝐹 ∈ V ∧ 𝑍𝑊) → (𝐹 supp 𝑍) = {𝑖 ∈ dom 𝐹 ∣ (𝐹𝑖) ≠ 𝑍})
72, 4, 5, 6syl3anc 1373 . 2 ((𝐹 Fn 𝑋𝑋𝑉𝑍𝑊) → (𝐹 supp 𝑍) = {𝑖 ∈ dom 𝐹 ∣ (𝐹𝑖) ≠ 𝑍})
8 fndm 6671 . . . 4 (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
983ad2ant1 1134 . . 3 ((𝐹 Fn 𝑋𝑋𝑉𝑍𝑊) → dom 𝐹 = 𝑋)
109rabeqdv 3452 . 2 ((𝐹 Fn 𝑋𝑋𝑉𝑍𝑊) → {𝑖 ∈ dom 𝐹 ∣ (𝐹𝑖) ≠ 𝑍} = {𝑖𝑋 ∣ (𝐹𝑖) ≠ 𝑍})
117, 10eqtrd 2777 1 ((𝐹 Fn 𝑋𝑋𝑉𝑍𝑊) → (𝐹 supp 𝑍) = {𝑖𝑋 ∣ (𝐹𝑖) ≠ 𝑍})
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1540  wcel 2108  wne 2940  {crab 3436  Vcvv 3480  dom cdm 5685  Fun wfun 6555   Fn wfn 6556  cfv 6561  (class class class)co 7431   supp csupp 8185
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-supp 8186
This theorem is referenced by:  elsuppfn  8195  cantnflem1  9729  fsuppmapnn0fiub0  14034  fsuppmapnn0ub  14036  mptnn0fsupp  14038  mptnn0fsuppr  14040  cicer  17850  rrgsupp  20701  mptscmfsupp0  20925  frlmbas  21775  frlmssuvc2  21815  pmatcollpw2lem  22783  rrxmvallem  25438  fpwrelmapffslem  32743  fedgmullem2  33681  fsumcvg4  33949  fsuppind  42600  fsumsupp0  45593
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