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Theorem suppval 7634
 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 7633 . . 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 5619 . . . . 5 (𝑥 = 𝑋 → dom 𝑥 = dom 𝑋)
43adantr 473 . . . 4 ((𝑥 = 𝑋𝑧 = 𝑍) → dom 𝑥 = dom 𝑋)
5 imaeq1 5763 . . . . . 6 (𝑥 = 𝑋 → (𝑥 “ {𝑖}) = (𝑋 “ {𝑖}))
65adantr 473 . . . . 5 ((𝑥 = 𝑋𝑧 = 𝑍) → (𝑥 “ {𝑖}) = (𝑋 “ {𝑖}))
7 sneq 4446 . . . . . 6 (𝑧 = 𝑍 → {𝑧} = {𝑍})
87adantl 474 . . . . 5 ((𝑥 = 𝑋𝑧 = 𝑍) → {𝑧} = {𝑍})
96, 8neeq12d 3023 . . . 4 ((𝑥 = 𝑋𝑧 = 𝑍) → ((𝑥 “ {𝑖}) ≠ {𝑧} ↔ (𝑋 “ {𝑖}) ≠ {𝑍}))
104, 9rabeqbidv 3403 . . 3 ((𝑥 = 𝑋𝑧 = 𝑍) → {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}} = {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}})
1110adantl 474 . 2 (((𝑋𝑉𝑍𝑊) ∧ (𝑥 = 𝑋𝑧 = 𝑍)) → {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}} = {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}})
12 elex 3428 . . 3 (𝑋𝑉𝑋 ∈ V)
1312adantr 473 . 2 ((𝑋𝑉𝑍𝑊) → 𝑋 ∈ V)
14 elex 3428 . . 3 (𝑍𝑊𝑍 ∈ V)
1514adantl 474 . 2 ((𝑋𝑉𝑍𝑊) → 𝑍 ∈ V)
16 dmexg 7427 . . . 4 (𝑋𝑉 → dom 𝑋 ∈ V)
1716adantr 473 . . 3 ((𝑋𝑉𝑍𝑊) → dom 𝑋 ∈ V)
18 rabexg 5087 . . 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 7117 1 ((𝑋𝑉𝑍𝑊) → (𝑋 supp 𝑍) = {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 387   = wceq 1508   ∈ wcel 2051   ≠ wne 2962  {crab 3087  Vcvv 3410  {csn 4436  dom cdm 5404   “ cima 5407  (class class class)co 6975   ∈ cmpo 6977   supp csupp 7632 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-sep 5057  ax-nul 5064  ax-pr 5183  ax-un 7278 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ne 2963  df-ral 3088  df-rex 3089  df-rab 3092  df-v 3412  df-sbc 3677  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-nul 4174  df-if 4346  df-sn 4437  df-pr 4439  df-op 4443  df-uni 4710  df-br 4927  df-opab 4989  df-id 5309  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-iota 6150  df-fun 6188  df-fv 6194  df-ov 6978  df-oprab 6979  df-mpo 6980  df-supp 7633 This theorem is referenced by:  suppvalbr  7636  supp0  7637  suppval1  7638  suppssdm  7645  suppsnop  7646  ressuppss  7651  ressuppssdif  7653
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