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Theorem suppval1 7536
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 7532 . . 3 ((𝑋𝑉𝑍𝑊) → (𝑋 supp 𝑍) = {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}})
213adant1 1161 . 2 ((Fun 𝑋𝑋𝑉𝑍𝑊) → (𝑋 supp 𝑍) = {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}})
3 funfn 6129 . . . . . . . . 9 (Fun 𝑋𝑋 Fn dom 𝑋)
43biimpi 208 . . . . . . . 8 (Fun 𝑋𝑋 Fn dom 𝑋)
543ad2ant1 1164 . . . . . . 7 ((Fun 𝑋𝑋𝑉𝑍𝑊) → 𝑋 Fn dom 𝑋)
6 fnsnfv 6481 . . . . . . 7 ((𝑋 Fn dom 𝑋𝑖 ∈ dom 𝑋) → {(𝑋𝑖)} = (𝑋 “ {𝑖}))
75, 6sylan 576 . . . . . 6 (((Fun 𝑋𝑋𝑉𝑍𝑊) ∧ 𝑖 ∈ dom 𝑋) → {(𝑋𝑖)} = (𝑋 “ {𝑖}))
87eqcomd 2803 . . . . 5 (((Fun 𝑋𝑋𝑉𝑍𝑊) ∧ 𝑖 ∈ dom 𝑋) → (𝑋 “ {𝑖}) = {(𝑋𝑖)})
98neeq1d 3028 . . . 4 (((Fun 𝑋𝑋𝑉𝑍𝑊) ∧ 𝑖 ∈ dom 𝑋) → ((𝑋 “ {𝑖}) ≠ {𝑍} ↔ {(𝑋𝑖)} ≠ {𝑍}))
10 fvex 6422 . . . . . 6 (𝑋𝑖) ∈ V
11 sneqbg 4558 . . . . . 6 ((𝑋𝑖) ∈ V → ({(𝑋𝑖)} = {𝑍} ↔ (𝑋𝑖) = 𝑍))
1210, 11mp1i 13 . . . . 5 (((Fun 𝑋𝑋𝑉𝑍𝑊) ∧ 𝑖 ∈ dom 𝑋) → ({(𝑋𝑖)} = {𝑍} ↔ (𝑋𝑖) = 𝑍))
1312necon3bid 3013 . . . 4 (((Fun 𝑋𝑋𝑉𝑍𝑊) ∧ 𝑖 ∈ dom 𝑋) → ({(𝑋𝑖)} ≠ {𝑍} ↔ (𝑋𝑖) ≠ 𝑍))
149, 13bitrd 271 . . 3 (((Fun 𝑋𝑋𝑉𝑍𝑊) ∧ 𝑖 ∈ dom 𝑋) → ((𝑋 “ {𝑖}) ≠ {𝑍} ↔ (𝑋𝑖) ≠ 𝑍))
1514rabbidva 3370 . 2 ((Fun 𝑋𝑋𝑉𝑍𝑊) → {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}} = {𝑖 ∈ dom 𝑋 ∣ (𝑋𝑖) ≠ 𝑍})
162, 15eqtrd 2831 1 ((Fun 𝑋𝑋𝑉𝑍𝑊) → (𝑋 supp 𝑍) = {𝑖 ∈ dom 𝑋 ∣ (𝑋𝑖) ≠ 𝑍})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108   = wceq 1653  wcel 2157  wne 2969  {crab 3091  Vcvv 3383  {csn 4366  dom cdm 5310  cima 5313  Fun wfun 6093   Fn wfn 6094  cfv 6099  (class class class)co 6876   supp csupp 7530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-br 4842  df-opab 4904  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-fv 6107  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-supp 7531
This theorem is referenced by:  suppvalfn  7537  suppfnss  7555  suppfnssOLD  7556  fnsuppres  7558  domnmsuppn0  42937  rmsuppss  42938  mndpsuppss  42939  scmsuppss  42940  suppdm  43087
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