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Theorem suppvalfng 8164
Description: The value of the operation constructing the support of a function with a given domain. This version of suppvalfn 8165 assumes 𝐹 is a set rather than its domain 𝑋, avoiding ax-rep 5249. (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 6637 . . 3 (𝐹 Fn 𝑋 → Fun 𝐹)
2 suppval1 8163 . . 3 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝐹 supp 𝑍) = {𝑖 ∈ dom 𝐹 ∣ (𝐹𝑖) ≠ 𝑍})
31, 2syl3an1 1163 . 2 ((𝐹 Fn 𝑋𝐹𝑉𝑍𝑊) → (𝐹 supp 𝑍) = {𝑖 ∈ dom 𝐹 ∣ (𝐹𝑖) ≠ 𝑍})
4 fndm 6640 . . . 4 (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
543ad2ant1 1133 . . 3 ((𝐹 Fn 𝑋𝐹𝑉𝑍𝑊) → dom 𝐹 = 𝑋)
65rabeqdv 3431 . 2 ((𝐹 Fn 𝑋𝐹𝑉𝑍𝑊) → {𝑖 ∈ dom 𝐹 ∣ (𝐹𝑖) ≠ 𝑍} = {𝑖𝑋 ∣ (𝐹𝑖) ≠ 𝑍})
73, 6eqtrd 2770 1 ((𝐹 Fn 𝑋𝐹𝑉𝑍𝑊) → (𝐹 supp 𝑍) = {𝑖𝑋 ∣ (𝐹𝑖) ≠ 𝑍})
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1540  wcel 2108  wne 2932  {crab 3415  dom cdm 5654  Fun wfun 6524   Fn wfn 6525  cfv 6530  (class class class)co 7403   supp csupp 8157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7727
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6483  df-fun 6532  df-fn 6533  df-fv 6538  df-ov 7406  df-oprab 7407  df-mpo 7408  df-supp 8158
This theorem is referenced by:  elsuppfng  8166
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