Theorem suppval 8167

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 8166	. . 3 ⊢ supp = (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}})
2	1	a1i 11	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → supp = (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}}))
3		dmeq 5898	. . . . 5 ⊢ (𝑥 = 𝑋 → dom 𝑥 = dom 𝑋)
4	3	adantr 486	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑧 = 𝑍) → dom 𝑥 = dom 𝑋)
5		imaeq1 6062	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 “ {𝑖}) = (𝑋 “ {𝑖}))
6	5	adantr 486	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑧 = 𝑍) → (𝑥 “ {𝑖}) = (𝑋 “ {𝑖}))
7		sneq 4604	. . . . . 6 ⊢ (𝑧 = 𝑍 → {𝑧} = {𝑍})
8	7	adantl 487	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑧 = 𝑍) → {𝑧} = {𝑍})
9	6, 8	neeq12d 3022	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑧 = 𝑍) → ((𝑥 “ {𝑖}) ≠ {𝑧} ↔ (𝑋 “ {𝑖}) ≠ {𝑍}))
10	4, 9	rabeqbidv 3437	. . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑧 = 𝑍) → {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}} = {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}})
11	10	adantl 487	. 2 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ (𝑥 = 𝑋 ∧ 𝑧 = 𝑍)) → {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}} = {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}})
12		elex 3479	. . 3 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V)
13	12	adantr 486	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → 𝑋 ∈ V)
14		elex 3479	. . 3 ⊢ (𝑍 ∈ 𝑊 → 𝑍 ∈ V)
15	14	adantl 487	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → 𝑍 ∈ V)
16		dmexg 7907	. . . 4 ⊢ (𝑋 ∈ 𝑉 → dom 𝑋 ∈ V)
17	16	adantr 486	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → dom 𝑋 ∈ V)
18		rabexg 5313	. . 3 ⊢ (dom 𝑋 ∈ V → {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}} ∈ V)
19	17, 18	syl 18	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}} ∈ V)
20	2, 11, 13, 15, 19	ovmpod 7575	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑋 supp 𝑍) = {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ≠ wne 2961  {crab 3419  Vcvv 3458  {csn 4594  dom cdm 5666   “ cima 5669  (class class class)co 7423   ∈ cmpo 7425   supp csupp 8165

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpo 7428  df-supp 8166

This theorem is used by:  suppvalbr  8169  supp0  8170  suppval1  8171  suppssdm  8182  suppsnop  8183  ressuppss  8188  ressuppssdif  8190