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Theorem suppvalfn 8191
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 1132 . . 3 ((𝐹 Fn 𝑋𝑋𝑉𝑍𝑊) → Fun 𝐹)
3 fnex 7236 . . . 4 ((𝐹 Fn 𝑋𝑋𝑉) → 𝐹 ∈ V)
433adant3 1131 . . 3 ((𝐹 Fn 𝑋𝑋𝑉𝑍𝑊) → 𝐹 ∈ V)
5 simp3 1137 . . 3 ((𝐹 Fn 𝑋𝑋𝑉𝑍𝑊) → 𝑍𝑊)
6 suppval1 8189 . . 3 ((Fun 𝐹𝐹 ∈ V ∧ 𝑍𝑊) → (𝐹 supp 𝑍) = {𝑖 ∈ dom 𝐹 ∣ (𝐹𝑖) ≠ 𝑍})
72, 4, 5, 6syl3anc 1370 . 2 ((𝐹 Fn 𝑋𝑋𝑉𝑍𝑊) → (𝐹 supp 𝑍) = {𝑖 ∈ dom 𝐹 ∣ (𝐹𝑖) ≠ 𝑍})
8 fndm 6671 . . . 4 (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
983ad2ant1 1132 . . 3 ((𝐹 Fn 𝑋𝑋𝑉𝑍𝑊) → dom 𝐹 = 𝑋)
109rabeqdv 3448 . 2 ((𝐹 Fn 𝑋𝑋𝑉𝑍𝑊) → {𝑖 ∈ dom 𝐹 ∣ (𝐹𝑖) ≠ 𝑍} = {𝑖𝑋 ∣ (𝐹𝑖) ≠ 𝑍})
117, 10eqtrd 2774 1 ((𝐹 Fn 𝑋𝑋𝑉𝑍𝑊) → (𝐹 supp 𝑍) = {𝑖𝑋 ∣ (𝐹𝑖) ≠ 𝑍})
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1536  wcel 2105  wne 2937  {crab 3432  Vcvv 3477  dom cdm 5688  Fun wfun 6556   Fn wfn 6557  cfv 6562  (class class class)co 7430   supp csupp 8183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-supp 8184
This theorem is referenced by:  elsuppfn  8193  cantnflem1  9726  fsuppmapnn0fiub0  14030  fsuppmapnn0ub  14032  mptnn0fsupp  14034  mptnn0fsuppr  14036  cicer  17853  rrgsupp  20717  mptscmfsupp0  20941  frlmbas  21792  frlmssuvc2  21832  pmatcollpw2lem  22798  rrxmvallem  25451  fpwrelmapffslem  32749  fedgmullem2  33657  fsumcvg4  33910  fsuppind  42576  fsumsupp0  45533
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