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Theorem suppvalfng 8153
Description: The value of the operation constructing the support of a function with a given domain. This version of suppvalfn 8154 assumes 𝐹 is a set rather than its domain 𝑋, avoiding ax-rep 5278. (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 6643 . . 3 (𝐹 Fn 𝑋 → Fun 𝐹)
2 suppval1 8152 . . 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 6646 . . . 4 (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
543ad2ant1 1130 . . 3 ((𝐹 Fn 𝑋𝐹𝑉𝑍𝑊) → dom 𝐹 = 𝑋)
65rabeqdv 3441 . 2 ((𝐹 Fn 𝑋𝐹𝑉𝑍𝑊) → {𝑖 ∈ dom 𝐹 ∣ (𝐹𝑖) ≠ 𝑍} = {𝑖𝑋 ∣ (𝐹𝑖) ≠ 𝑍})
73, 6eqtrd 2766 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 2934  {crab 3426  dom cdm 5669  Fun wfun 6531   Fn wfn 6532  cfv 6537  (class class class)co 7405   supp csupp 8146
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-supp 8147
This theorem is referenced by:  elsuppfng  8155
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