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

Theorem suppval 8187
Description: The value of the operation constructing the support of a function. (Contributed by AV, 31-Mar-2019.) (Revised by AV, 6-Apr-2019.)
Assertion
Ref Expression
suppval ((𝑋𝑉𝑍𝑊) → (𝑋 supp 𝑍) = {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}})
Distinct variable groups:   𝑖,𝑋   𝑖,𝑍
Allowed substitution hints:   𝑉(𝑖)   𝑊(𝑖)

Proof of Theorem suppval
Dummy variables 𝑥 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-supp 8186 . . 3 supp = (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}})
21a1i 11 . 2 ((𝑋𝑉𝑍𝑊) → supp = (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}}))
3 dmeq 5914 . . . . 5 (𝑥 = 𝑋 → dom 𝑥 = dom 𝑋)
43adantr 480 . . . 4 ((𝑥 = 𝑋𝑧 = 𝑍) → dom 𝑥 = dom 𝑋)
5 imaeq1 6073 . . . . . 6 (𝑥 = 𝑋 → (𝑥 “ {𝑖}) = (𝑋 “ {𝑖}))
65adantr 480 . . . . 5 ((𝑥 = 𝑋𝑧 = 𝑍) → (𝑥 “ {𝑖}) = (𝑋 “ {𝑖}))
7 sneq 4636 . . . . . 6 (𝑧 = 𝑍 → {𝑧} = {𝑍})
87adantl 481 . . . . 5 ((𝑥 = 𝑋𝑧 = 𝑍) → {𝑧} = {𝑍})
96, 8neeq12d 3002 . . . 4 ((𝑥 = 𝑋𝑧 = 𝑍) → ((𝑥 “ {𝑖}) ≠ {𝑧} ↔ (𝑋 “ {𝑖}) ≠ {𝑍}))
104, 9rabeqbidv 3455 . . 3 ((𝑥 = 𝑋𝑧 = 𝑍) → {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}} = {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}})
1110adantl 481 . 2 (((𝑋𝑉𝑍𝑊) ∧ (𝑥 = 𝑋𝑧 = 𝑍)) → {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}} = {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}})
12 elex 3501 . . 3 (𝑋𝑉𝑋 ∈ V)
1312adantr 480 . 2 ((𝑋𝑉𝑍𝑊) → 𝑋 ∈ V)
14 elex 3501 . . 3 (𝑍𝑊𝑍 ∈ V)
1514adantl 481 . 2 ((𝑋𝑉𝑍𝑊) → 𝑍 ∈ V)
16 dmexg 7923 . . . 4 (𝑋𝑉 → dom 𝑋 ∈ V)
1716adantr 480 . . 3 ((𝑋𝑉𝑍𝑊) → dom 𝑋 ∈ V)
18 rabexg 5337 . . 3 (dom 𝑋 ∈ V → {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}} ∈ V)
1917, 18syl 17 . 2 ((𝑋𝑉𝑍𝑊) → {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}} ∈ V)
202, 11, 13, 15, 19ovmpod 7585 1 ((𝑋𝑉𝑍𝑊) → (𝑋 supp 𝑍) = {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wne 2940  {crab 3436  Vcvv 3480  {csn 4626  dom cdm 5685  cima 5688  (class class class)co 7431  cmpo 7433   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-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-supp 8186
This theorem is referenced by:  suppvalbr  8189  supp0  8190  suppval1  8191  suppssdm  8202  suppsnop  8203  ressuppss  8208  ressuppssdif  8210
  Copyright terms: Public domain W3C validator