MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elsuppfng Structured version   Visualization version   GIF version

Theorem elsuppfng 8194
Description: An element of the support of a function with a given domain. This version of elsuppfn 8195 assumes 𝐹 is a set rather than its domain 𝑋, avoiding ax-rep 5279. (Contributed by SN, 5-Aug-2024.)
Assertion
Ref Expression
elsuppfng ((𝐹 Fn 𝑋𝐹𝑉𝑍𝑊) → (𝑆 ∈ (𝐹 supp 𝑍) ↔ (𝑆𝑋 ∧ (𝐹𝑆) ≠ 𝑍)))

Proof of Theorem elsuppfng
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 suppvalfng 8192 . . 3 ((𝐹 Fn 𝑋𝐹𝑉𝑍𝑊) → (𝐹 supp 𝑍) = {𝑖𝑋 ∣ (𝐹𝑖) ≠ 𝑍})
21eleq2d 2827 . 2 ((𝐹 Fn 𝑋𝐹𝑉𝑍𝑊) → (𝑆 ∈ (𝐹 supp 𝑍) ↔ 𝑆 ∈ {𝑖𝑋 ∣ (𝐹𝑖) ≠ 𝑍}))
3 fveq2 6906 . . . 4 (𝑖 = 𝑆 → (𝐹𝑖) = (𝐹𝑆))
43neeq1d 3000 . . 3 (𝑖 = 𝑆 → ((𝐹𝑖) ≠ 𝑍 ↔ (𝐹𝑆) ≠ 𝑍))
54elrab 3692 . 2 (𝑆 ∈ {𝑖𝑋 ∣ (𝐹𝑖) ≠ 𝑍} ↔ (𝑆𝑋 ∧ (𝐹𝑆) ≠ 𝑍))
62, 5bitrdi 287 1 ((𝐹 Fn 𝑋𝐹𝑉𝑍𝑊) → (𝑆 ∈ (𝐹 supp 𝑍) ↔ (𝑆𝑋 ∧ (𝐹𝑆) ≠ 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  {crab 3436   Fn wfn 6556  cfv 6561  (class class class)co 7431   supp csupp 8185
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 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-supp 8186
This theorem is referenced by:  suppss  8219  suppssrg  8221  ciclcl  17846  cicrcl  17847  ismhp3  22146  mdegleb  26103  suppiniseg  32695
  Copyright terms: Public domain W3C validator