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Theorem suppvalfn 7688
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 6323 . . . 4 (𝐹 Fn 𝑋 → Fun 𝐹)
213ad2ant1 1126 . . 3 ((𝐹 Fn 𝑋𝑋𝑉𝑍𝑊) → Fun 𝐹)
3 fnex 6846 . . . 4 ((𝐹 Fn 𝑋𝑋𝑉) → 𝐹 ∈ V)
433adant3 1125 . . 3 ((𝐹 Fn 𝑋𝑋𝑉𝑍𝑊) → 𝐹 ∈ V)
5 simp3 1131 . . 3 ((𝐹 Fn 𝑋𝑋𝑉𝑍𝑊) → 𝑍𝑊)
6 suppval1 7687 . . 3 ((Fun 𝐹𝐹 ∈ V ∧ 𝑍𝑊) → (𝐹 supp 𝑍) = {𝑖 ∈ dom 𝐹 ∣ (𝐹𝑖) ≠ 𝑍})
72, 4, 5, 6syl3anc 1364 . 2 ((𝐹 Fn 𝑋𝑋𝑉𝑍𝑊) → (𝐹 supp 𝑍) = {𝑖 ∈ dom 𝐹 ∣ (𝐹𝑖) ≠ 𝑍})
8 fndm 6325 . . . 4 (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
983ad2ant1 1126 . . 3 ((𝐹 Fn 𝑋𝑋𝑉𝑍𝑊) → dom 𝐹 = 𝑋)
109rabeqdv 3429 . 2 ((𝐹 Fn 𝑋𝑋𝑉𝑍𝑊) → {𝑖 ∈ dom 𝐹 ∣ (𝐹𝑖) ≠ 𝑍} = {𝑖𝑋 ∣ (𝐹𝑖) ≠ 𝑍})
117, 10eqtrd 2831 1 ((𝐹 Fn 𝑋𝑋𝑉𝑍𝑊) → (𝐹 supp 𝑍) = {𝑖𝑋 ∣ (𝐹𝑖) ≠ 𝑍})
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1080   = wceq 1522  wcel 2081  wne 2984  {crab 3109  Vcvv 3437  dom cdm 5443  Fun wfun 6219   Fn wfn 6220  cfv 6225  (class class class)co 7016   supp csupp 7681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-ov 7019  df-oprab 7020  df-mpo 7021  df-supp 7682
This theorem is referenced by:  elsuppfn  7689  cantnflem1  8998  fsuppmapnn0fiub0  13211  fsuppmapnn0ub  13213  mptnn0fsupp  13215  mptnn0fsuppr  13217  cicer  16905  mptscmfsupp0  19389  rrgsupp  19753  mhpinvcl  20022  frlmbas  20581  frlmssuvc2  20621  pmatcollpw2lem  21069  rrxmvallem  23690  fpwrelmapffslem  30156  fedgmullem2  30630  fsumcvg4  30810  fsumsupp0  41401
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