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Theorem suppval 8186
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 8185 . . 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 5917 . . . . 5 (𝑥 = 𝑋 → dom 𝑥 = dom 𝑋)
43adantr 480 . . . 4 ((𝑥 = 𝑋𝑧 = 𝑍) → dom 𝑥 = dom 𝑋)
5 imaeq1 6075 . . . . . 6 (𝑥 = 𝑋 → (𝑥 “ {𝑖}) = (𝑋 “ {𝑖}))
65adantr 480 . . . . 5 ((𝑥 = 𝑋𝑧 = 𝑍) → (𝑥 “ {𝑖}) = (𝑋 “ {𝑖}))
7 sneq 4641 . . . . . 6 (𝑧 = 𝑍 → {𝑧} = {𝑍})
87adantl 481 . . . . 5 ((𝑥 = 𝑋𝑧 = 𝑍) → {𝑧} = {𝑍})
96, 8neeq12d 3000 . . . 4 ((𝑥 = 𝑋𝑧 = 𝑍) → ((𝑥 “ {𝑖}) ≠ {𝑧} ↔ (𝑋 “ {𝑖}) ≠ {𝑍}))
104, 9rabeqbidv 3452 . . 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 3499 . . 3 (𝑋𝑉𝑋 ∈ V)
1312adantr 480 . 2 ((𝑋𝑉𝑍𝑊) → 𝑋 ∈ V)
14 elex 3499 . . 3 (𝑍𝑊𝑍 ∈ V)
1514adantl 481 . 2 ((𝑋𝑉𝑍𝑊) → 𝑍 ∈ V)
16 dmexg 7924 . . . 4 (𝑋𝑉 → dom 𝑋 ∈ V)
1716adantr 480 . . 3 ((𝑋𝑉𝑍𝑊) → dom 𝑋 ∈ V)
18 rabexg 5343 . . 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 1537  wcel 2106  wne 2938  {crab 3433  Vcvv 3478  {csn 4631  dom cdm 5689  cima 5692  (class class class)co 7431  cmpo 7433   supp csupp 8184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-supp 8185
This theorem is referenced by:  suppvalbr  8188  supp0  8189  suppval1  8190  suppssdm  8201  suppsnop  8202  ressuppss  8207  ressuppssdif  8209
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