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Theorem suppvalfng 8100
Description: The value of the operation constructing the support of a function with a given domain. This version of suppvalfn 8101 assumes 𝐹 is a set rather than its domain 𝑋, avoiding ax-rep 5243. (Contributed by SN, 5-Aug-2024.)
Assertion
Ref Expression
suppvalfng ((𝐹 Fn 𝑋𝐹𝑉𝑍𝑊) → (𝐹 supp 𝑍) = {𝑖𝑋 ∣ (𝐹𝑖) ≠ 𝑍})
Distinct variable groups:   𝑖,𝑉   𝑖,𝑊   𝑖,𝑋   𝑖,𝑍   𝑖,𝐹

Proof of Theorem suppvalfng
StepHypRef Expression
1 fnfun 6603 . . 3 (𝐹 Fn 𝑋 → Fun 𝐹)
2 suppval1 8099 . . 3 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝐹 supp 𝑍) = {𝑖 ∈ dom 𝐹 ∣ (𝐹𝑖) ≠ 𝑍})
31, 2syl3an1 1164 . 2 ((𝐹 Fn 𝑋𝐹𝑉𝑍𝑊) → (𝐹 supp 𝑍) = {𝑖 ∈ dom 𝐹 ∣ (𝐹𝑖) ≠ 𝑍})
4 fndm 6606 . . . 4 (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
543ad2ant1 1134 . . 3 ((𝐹 Fn 𝑋𝐹𝑉𝑍𝑊) → dom 𝐹 = 𝑋)
65rabeqdv 3423 . 2 ((𝐹 Fn 𝑋𝐹𝑉𝑍𝑊) → {𝑖 ∈ dom 𝐹 ∣ (𝐹𝑖) ≠ 𝑍} = {𝑖𝑋 ∣ (𝐹𝑖) ≠ 𝑍})
73, 6eqtrd 2777 1 ((𝐹 Fn 𝑋𝐹𝑉𝑍𝑊) → (𝐹 supp 𝑍) = {𝑖𝑋 ∣ (𝐹𝑖) ≠ 𝑍})
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1088   = wceq 1542  wcel 2107  wne 2944  {crab 3408  dom cdm 5634  Fun wfun 6491   Fn wfn 6492  cfv 6497  (class class class)co 7358   supp csupp 8093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-supp 8094
This theorem is referenced by:  elsuppfng  8102
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