Theorem suppval 8151

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 8150	. . 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 5903	. . . . 5 ⊢ (𝑥 = 𝑋 → dom 𝑥 = dom 𝑋)
4	3	adantr 480	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑧 = 𝑍) → dom 𝑥 = dom 𝑋)
5		imaeq1 6054	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 “ {𝑖}) = (𝑋 “ {𝑖}))
6	5	adantr 480	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑧 = 𝑍) → (𝑥 “ {𝑖}) = (𝑋 “ {𝑖}))
7		sneq 4638	. . . . . 6 ⊢ (𝑧 = 𝑍 → {𝑧} = {𝑍})
8	7	adantl 481	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑧 = 𝑍) → {𝑧} = {𝑍})
9	6, 8	neeq12d 3001	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑧 = 𝑍) → ((𝑥 “ {𝑖}) ≠ {𝑧} ↔ (𝑋 “ {𝑖}) ≠ {𝑍}))
10	4, 9	rabeqbidv 3448	. . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑧 = 𝑍) → {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}} = {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}})
11	10	adantl 481	. 2 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ (𝑥 = 𝑋 ∧ 𝑧 = 𝑍)) → {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}} = {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}})
12		elex 3492	. . 3 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V)
13	12	adantr 480	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → 𝑋 ∈ V)
14		elex 3492	. . 3 ⊢ (𝑍 ∈ 𝑊 → 𝑍 ∈ V)
15	14	adantl 481	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → 𝑍 ∈ V)
16		dmexg 7897	. . . 4 ⊢ (𝑋 ∈ 𝑉 → dom 𝑋 ∈ V)
17	16	adantr 480	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → dom 𝑋 ∈ V)
18		rabexg 5331	. . 3 ⊢ (dom 𝑋 ∈ V → {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}} ∈ V)
19	17, 18	syl 17	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}} ∈ V)
20	2, 11, 13, 15, 19	ovmpod 7563	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑋 supp 𝑍) = {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105   ≠ wne 2939  {crab 3431  Vcvv 3473  {csn 4628  dom cdm 5676   “ cima 5679  (class class class)co 7412   ∈ cmpo 7414   supp csupp 8149

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7728

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-supp 8150

This theorem is referenced by:  suppvalbr  8153  supp0  8154  suppval1  8155  suppssdm  8165  suppsnop  8166  ressuppss  8171  ressuppssdif  8173