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Theorem suppval 7819
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 7818 . . 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 5749 . . . . 5 (𝑥 = 𝑋 → dom 𝑥 = dom 𝑋)
43adantr 484 . . . 4 ((𝑥 = 𝑋𝑧 = 𝑍) → dom 𝑥 = dom 𝑋)
5 imaeq1 5902 . . . . . 6 (𝑥 = 𝑋 → (𝑥 “ {𝑖}) = (𝑋 “ {𝑖}))
65adantr 484 . . . . 5 ((𝑥 = 𝑋𝑧 = 𝑍) → (𝑥 “ {𝑖}) = (𝑋 “ {𝑖}))
7 sneq 4549 . . . . . 6 (𝑧 = 𝑍 → {𝑧} = {𝑍})
87adantl 485 . . . . 5 ((𝑥 = 𝑋𝑧 = 𝑍) → {𝑧} = {𝑍})
96, 8neeq12d 3072 . . . 4 ((𝑥 = 𝑋𝑧 = 𝑍) → ((𝑥 “ {𝑖}) ≠ {𝑧} ↔ (𝑋 “ {𝑖}) ≠ {𝑍}))
104, 9rabeqbidv 3461 . . 3 ((𝑥 = 𝑋𝑧 = 𝑍) → {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}} = {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}})
1110adantl 485 . 2 (((𝑋𝑉𝑍𝑊) ∧ (𝑥 = 𝑋𝑧 = 𝑍)) → {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}} = {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}})
12 elex 3487 . . 3 (𝑋𝑉𝑋 ∈ V)
1312adantr 484 . 2 ((𝑋𝑉𝑍𝑊) → 𝑋 ∈ V)
14 elex 3487 . . 3 (𝑍𝑊𝑍 ∈ V)
1514adantl 485 . 2 ((𝑋𝑉𝑍𝑊) → 𝑍 ∈ V)
16 dmexg 7599 . . . 4 (𝑋𝑉 → dom 𝑋 ∈ V)
1716adantr 484 . . 3 ((𝑋𝑉𝑍𝑊) → dom 𝑋 ∈ V)
18 rabexg 5210 . . 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 7286 1 ((𝑋𝑉𝑍𝑊) → (𝑋 supp 𝑍) = {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  wne 3011  {crab 3134  Vcvv 3469  {csn 4539  dom cdm 5532  cima 5535  (class class class)co 7140  cmpo 7142   supp csupp 7817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-supp 7818
This theorem is referenced by:  suppvalbr  7821  supp0  7822  suppval1  7823  suppssdm  7830  suppsnop  7831  ressuppss  7836  ressuppssdif  7838
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