Theorem suppval 6481

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	⊢ ((𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (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.

Step	Hyp	Ref	Expression
1		df-supp 6480	. . 3 ⊢ supp = (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}})
2	1	a1i 9	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → supp = (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}}))
3		dmeq 4983	. . . . 5 ⊢ (𝑥 = 𝑋 → dom 𝑥 = dom 𝑋)
4	3	adantr 276	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑧 = 𝑍) → dom 𝑥 = dom 𝑋)
5		imaeq1 5124	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 “ {𝑖}) = (𝑋 “ {𝑖}))
6	5	adantr 276	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑧 = 𝑍) → (𝑥 “ {𝑖}) = (𝑋 “ {𝑖}))
7		sneq 3720	. . . . . 6 ⊢ (𝑧 = 𝑍 → {𝑧} = {𝑍})
8	7	adantl 277	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑧 = 𝑍) → {𝑧} = {𝑍})
9	6, 8	neeq12d 2440	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑧 = 𝑍) → ((𝑥 “ {𝑖}) ≠ {𝑧} ↔ (𝑋 “ {𝑖}) ≠ {𝑍}))
10	4, 9	rabeqbidv 2816	. . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑧 = 𝑍) → {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}} = {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}})
11	10	adantl 277	. 2 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ (𝑥 = 𝑋 ∧ 𝑧 = 𝑍)) → {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}} = {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}})
12		elex 2833	. . 3 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V)
13	12	adantr 276	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → 𝑋 ∈ V)
14		elex 2833	. . 3 ⊢ (𝑍 ∈ 𝑊 → 𝑍 ∈ V)
15	14	adantl 277	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → 𝑍 ∈ V)
16		dmexg 5048	. . . 4 ⊢ (𝑋 ∈ 𝑉 → dom 𝑋 ∈ V)
17	16	adantr 276	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → dom 𝑋 ∈ V)
18		rabexg 4281	. . 3 ⊢ (dom 𝑋 ∈ V → {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}} ∈ V)
19	17, 18	syl 14	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}} ∈ V)
20	2, 11, 13, 15, 19	ovmpod 6219	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑋 supp 𝑍) = {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}})

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209   ≠ wne 2420  {crab 2532  Vcvv 2821  {csn 3709  dom cdm 4776   “ cima 4779  (class class class)co 6088   ∈ cmpo 6090   supp csupp 6479

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4132  df-opab 4194  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-supp 6480

This theorem is used by:  supp0  6482  suppval1  6483  suppssdmg  6493  suppsnopdc  6494  ressuppss  6498