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Theorem suppvalfn 8171
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 6649 . . . 4 (𝐹 Fn 𝑋 → Fun 𝐹)
213ad2ant1 1130 . . 3 ((𝐹 Fn 𝑋𝑋𝑉𝑍𝑊) → Fun 𝐹)
3 fnex 7225 . . . 4 ((𝐹 Fn 𝑋𝑋𝑉) → 𝐹 ∈ V)
433adant3 1129 . . 3 ((𝐹 Fn 𝑋𝑋𝑉𝑍𝑊) → 𝐹 ∈ V)
5 simp3 1135 . . 3 ((𝐹 Fn 𝑋𝑋𝑉𝑍𝑊) → 𝑍𝑊)
6 suppval1 8169 . . 3 ((Fun 𝐹𝐹 ∈ V ∧ 𝑍𝑊) → (𝐹 supp 𝑍) = {𝑖 ∈ dom 𝐹 ∣ (𝐹𝑖) ≠ 𝑍})
72, 4, 5, 6syl3anc 1368 . 2 ((𝐹 Fn 𝑋𝑋𝑉𝑍𝑊) → (𝐹 supp 𝑍) = {𝑖 ∈ dom 𝐹 ∣ (𝐹𝑖) ≠ 𝑍})
8 fndm 6652 . . . 4 (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
983ad2ant1 1130 . . 3 ((𝐹 Fn 𝑋𝑋𝑉𝑍𝑊) → dom 𝐹 = 𝑋)
109rabeqdv 3435 . 2 ((𝐹 Fn 𝑋𝑋𝑉𝑍𝑊) → {𝑖 ∈ dom 𝐹 ∣ (𝐹𝑖) ≠ 𝑍} = {𝑖𝑋 ∣ (𝐹𝑖) ≠ 𝑍})
117, 10eqtrd 2765 1 ((𝐹 Fn 𝑋𝑋𝑉𝑍𝑊) → (𝐹 supp 𝑍) = {𝑖𝑋 ∣ (𝐹𝑖) ≠ 𝑍})
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1533  wcel 2098  wne 2930  {crab 3419  Vcvv 3463  dom cdm 5672  Fun wfun 6537   Fn wfn 6538  cfv 6543  (class class class)co 7416   supp csupp 8163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7419  df-oprab 7420  df-mpo 7421  df-supp 8164
This theorem is referenced by:  elsuppfn  8173  cantnflem1  9712  fsuppmapnn0fiub0  13990  fsuppmapnn0ub  13992  mptnn0fsupp  13994  mptnn0fsuppr  13996  cicer  17788  mptscmfsupp0  20814  rrgsupp  21242  frlmbas  21693  frlmssuvc2  21733  pmatcollpw2lem  22697  rrxmvallem  25350  fpwrelmapffslem  32559  fedgmullem2  33385  fsumcvg4  33608  fsuppind  41888  fsumsupp0  45029
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