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Theorem suppval 7448
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 7447 . . 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 5462 . . . . 5 (𝑥 = 𝑋 → dom 𝑥 = dom 𝑋)
43adantr 466 . . . 4 ((𝑥 = 𝑋𝑧 = 𝑍) → dom 𝑥 = dom 𝑋)
5 imaeq1 5602 . . . . . 6 (𝑥 = 𝑋 → (𝑥 “ {𝑖}) = (𝑋 “ {𝑖}))
65adantr 466 . . . . 5 ((𝑥 = 𝑋𝑧 = 𝑍) → (𝑥 “ {𝑖}) = (𝑋 “ {𝑖}))
7 sneq 4326 . . . . . 6 (𝑧 = 𝑍 → {𝑧} = {𝑍})
87adantl 467 . . . . 5 ((𝑥 = 𝑋𝑧 = 𝑍) → {𝑧} = {𝑍})
96, 8neeq12d 3004 . . . 4 ((𝑥 = 𝑋𝑧 = 𝑍) → ((𝑥 “ {𝑖}) ≠ {𝑧} ↔ (𝑋 “ {𝑖}) ≠ {𝑍}))
104, 9rabeqbidv 3345 . . 3 ((𝑥 = 𝑋𝑧 = 𝑍) → {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}} = {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}})
1110adantl 467 . 2 (((𝑋𝑉𝑍𝑊) ∧ (𝑥 = 𝑋𝑧 = 𝑍)) → {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}} = {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}})
12 elex 3364 . . 3 (𝑋𝑉𝑋 ∈ V)
1312adantr 466 . 2 ((𝑋𝑉𝑍𝑊) → 𝑋 ∈ V)
14 elex 3364 . . 3 (𝑍𝑊𝑍 ∈ V)
1514adantl 467 . 2 ((𝑋𝑉𝑍𝑊) → 𝑍 ∈ V)
16 dmexg 7244 . . . 4 (𝑋𝑉 → dom 𝑋 ∈ V)
1716adantr 466 . . 3 ((𝑋𝑉𝑍𝑊) → dom 𝑋 ∈ V)
18 rabexg 4945 . . 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, 19ovmpt2d 6935 1 ((𝑋𝑉𝑍𝑊) → (𝑋 supp 𝑍) = {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wne 2943  {crab 3065  Vcvv 3351  {csn 4316  dom cdm 5249  cima 5252  (class class class)co 6793  cmpt2 6795   supp csupp 7446
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-supp 7447
This theorem is referenced by:  suppvalbr  7450  supp0  7451  suppval1  7452  suppssdm  7459  suppsnop  7460  ressuppss  7465  ressuppssdif  7467
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