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Theorem suppval1 7246
Description: The value of the operation constructing the support of a function. (Contributed by AV, 6-Apr-2019.)
Assertion
Ref Expression
suppval1 ((Fun 𝑋𝑋𝑉𝑍𝑊) → (𝑋 supp 𝑍) = {𝑖 ∈ dom 𝑋 ∣ (𝑋𝑖) ≠ 𝑍})
Distinct variable groups:   𝑖,𝑉   𝑖,𝑊   𝑖,𝑋   𝑖,𝑍

Proof of Theorem suppval1
StepHypRef Expression
1 suppval 7242 . . 3 ((𝑋𝑉𝑍𝑊) → (𝑋 supp 𝑍) = {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}})
213adant1 1077 . 2 ((Fun 𝑋𝑋𝑉𝑍𝑊) → (𝑋 supp 𝑍) = {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}})
3 funfn 5877 . . . . . . . . 9 (Fun 𝑋𝑋 Fn dom 𝑋)
43biimpi 206 . . . . . . . 8 (Fun 𝑋𝑋 Fn dom 𝑋)
543ad2ant1 1080 . . . . . . 7 ((Fun 𝑋𝑋𝑉𝑍𝑊) → 𝑋 Fn dom 𝑋)
6 fnsnfv 6215 . . . . . . 7 ((𝑋 Fn dom 𝑋𝑖 ∈ dom 𝑋) → {(𝑋𝑖)} = (𝑋 “ {𝑖}))
75, 6sylan 488 . . . . . 6 (((Fun 𝑋𝑋𝑉𝑍𝑊) ∧ 𝑖 ∈ dom 𝑋) → {(𝑋𝑖)} = (𝑋 “ {𝑖}))
87eqcomd 2627 . . . . 5 (((Fun 𝑋𝑋𝑉𝑍𝑊) ∧ 𝑖 ∈ dom 𝑋) → (𝑋 “ {𝑖}) = {(𝑋𝑖)})
98neeq1d 2849 . . . 4 (((Fun 𝑋𝑋𝑉𝑍𝑊) ∧ 𝑖 ∈ dom 𝑋) → ((𝑋 “ {𝑖}) ≠ {𝑍} ↔ {(𝑋𝑖)} ≠ {𝑍}))
10 fvex 6158 . . . . . 6 (𝑋𝑖) ∈ V
11 sneqbg 4342 . . . . . 6 ((𝑋𝑖) ∈ V → ({(𝑋𝑖)} = {𝑍} ↔ (𝑋𝑖) = 𝑍))
1210, 11mp1i 13 . . . . 5 (((Fun 𝑋𝑋𝑉𝑍𝑊) ∧ 𝑖 ∈ dom 𝑋) → ({(𝑋𝑖)} = {𝑍} ↔ (𝑋𝑖) = 𝑍))
1312necon3bid 2834 . . . 4 (((Fun 𝑋𝑋𝑉𝑍𝑊) ∧ 𝑖 ∈ dom 𝑋) → ({(𝑋𝑖)} ≠ {𝑍} ↔ (𝑋𝑖) ≠ 𝑍))
149, 13bitrd 268 . . 3 (((Fun 𝑋𝑋𝑉𝑍𝑊) ∧ 𝑖 ∈ dom 𝑋) → ((𝑋 “ {𝑖}) ≠ {𝑍} ↔ (𝑋𝑖) ≠ 𝑍))
1514rabbidva 3176 . 2 ((Fun 𝑋𝑋𝑉𝑍𝑊) → {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}} = {𝑖 ∈ dom 𝑋 ∣ (𝑋𝑖) ≠ 𝑍})
162, 15eqtrd 2655 1 ((Fun 𝑋𝑋𝑉𝑍𝑊) → (𝑋 supp 𝑍) = {𝑖 ∈ dom 𝑋 ∣ (𝑋𝑖) ≠ 𝑍})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  {crab 2911  Vcvv 3186  {csn 4148  dom cdm 5074  cima 5077  Fun wfun 5841   Fn wfn 5842  cfv 5847  (class class class)co 6604   supp csupp 7240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-supp 7241
This theorem is referenced by:  suppvalfn  7247  suppfnss  7265  fnsuppres  7267  domnmsuppn0  41438  rmsuppss  41439  mndpsuppss  41440  scmsuppss  41441  suppdm  41588
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