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Theorem suppvalfng 8165
Description: The value of the operation constructing the support of a function with a given domain. This version of suppvalfn 8166 assumes 𝐹 is a set rather than its domain 𝑋, avoiding ax-rep 5281. (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 6649 . . 3 (𝐹 Fn 𝑋 → Fun 𝐹)
2 suppval1 8164 . . 3 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝐹 supp 𝑍) = {𝑖 ∈ dom 𝐹 ∣ (𝐹𝑖) ≠ 𝑍})
31, 2syl3an1 1160 . 2 ((𝐹 Fn 𝑋𝐹𝑉𝑍𝑊) → (𝐹 supp 𝑍) = {𝑖 ∈ dom 𝐹 ∣ (𝐹𝑖) ≠ 𝑍})
4 fndm 6652 . . . 4 (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
543ad2ant1 1130 . . 3 ((𝐹 Fn 𝑋𝐹𝑉𝑍𝑊) → dom 𝐹 = 𝑋)
65rabeqdv 3435 . 2 ((𝐹 Fn 𝑋𝐹𝑉𝑍𝑊) → {𝑖 ∈ dom 𝐹 ∣ (𝐹𝑖) ≠ 𝑍} = {𝑖𝑋 ∣ (𝐹𝑖) ≠ 𝑍})
73, 6eqtrd 2765 1 ((𝐹 Fn 𝑋𝐹𝑉𝑍𝑊) → (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  dom cdm 5673  Fun wfun 6537   Fn wfn 6538  cfv 6543  (class class class)co 7413   supp csupp 8158
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-sep 5295  ax-nul 5302  ax-pr 5424  ax-un 7735
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-rab 3420  df-v 3465  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5145  df-opab 5207  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551  df-ov 7416  df-oprab 7417  df-mpo 7418  df-supp 8159
This theorem is referenced by:  elsuppfng  8167
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