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Theorem suppvalfn 7452
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 6128 . . . 4 (𝐹 Fn 𝑋 → Fun 𝐹)
213ad2ant1 1126 . . 3 ((𝐹 Fn 𝑋𝑋𝑉𝑍𝑊) → Fun 𝐹)
3 fnex 6624 . . . 4 ((𝐹 Fn 𝑋𝑋𝑉) → 𝐹 ∈ V)
433adant3 1125 . . 3 ((𝐹 Fn 𝑋𝑋𝑉𝑍𝑊) → 𝐹 ∈ V)
5 simp3 1131 . . 3 ((𝐹 Fn 𝑋𝑋𝑉𝑍𝑊) → 𝑍𝑊)
6 suppval1 7451 . . 3 ((Fun 𝐹𝐹 ∈ V ∧ 𝑍𝑊) → (𝐹 supp 𝑍) = {𝑖 ∈ dom 𝐹 ∣ (𝐹𝑖) ≠ 𝑍})
72, 4, 5, 6syl3anc 1475 . 2 ((𝐹 Fn 𝑋𝑋𝑉𝑍𝑊) → (𝐹 supp 𝑍) = {𝑖 ∈ dom 𝐹 ∣ (𝐹𝑖) ≠ 𝑍})
8 fndm 6130 . . . 4 (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
983ad2ant1 1126 . . 3 ((𝐹 Fn 𝑋𝑋𝑉𝑍𝑊) → dom 𝐹 = 𝑋)
10 rabeq 3341 . . 3 (dom 𝐹 = 𝑋 → {𝑖 ∈ dom 𝐹 ∣ (𝐹𝑖) ≠ 𝑍} = {𝑖𝑋 ∣ (𝐹𝑖) ≠ 𝑍})
119, 10syl 17 . 2 ((𝐹 Fn 𝑋𝑋𝑉𝑍𝑊) → {𝑖 ∈ dom 𝐹 ∣ (𝐹𝑖) ≠ 𝑍} = {𝑖𝑋 ∣ (𝐹𝑖) ≠ 𝑍})
127, 11eqtrd 2804 1 ((𝐹 Fn 𝑋𝑋𝑉𝑍𝑊) → (𝐹 supp 𝑍) = {𝑖𝑋 ∣ (𝐹𝑖) ≠ 𝑍})
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1070   = wceq 1630  wcel 2144  wne 2942  {crab 3064  Vcvv 3349  dom cdm 5249  Fun wfun 6025   Fn wfn 6026  cfv 6031  (class class class)co 6792   supp csupp 7445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-supp 7446
This theorem is referenced by:  elsuppfn  7453  cantnflem1  8749  fsuppmapnn0fiub0  12999  fsuppmapnn0ub  13001  mptnn0fsupp  13003  mptnn0fsuppr  13005  cicer  16672  mptscmfsupp0  19137  rrgsupp  19505  frlmbas  20315  frlmssuvc2  20350  pmatcollpw2lem  20801  rrxmvallem  23405  fpwrelmapffslem  29841  fsumcvg4  30330  fsumsupp0  40322
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