Theorem ressuppssdif 8001

Description: The support of the restriction of a function is a subset of the support of the function itself. (Contributed by AV, 22-Apr-2019.)

Assertion

Ref	Expression
ressuppssdif	⊢ ((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) ⊆ (((𝐹 ↾ 𝐵) supp 𝑍) ∪ (dom 𝐹 ∖ 𝐵)))

Proof of Theorem ressuppssdif

Dummy variables 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldif 3897	. . . . . 6 ⊢ (𝑥 ∈ ({𝑧 ∈ dom 𝐹 ∣ (𝐹 “ {𝑧}) ≠ {𝑍}} ∖ {𝑧 ∈ dom (𝐹 ↾ 𝐵) ∣ ((𝐹 ↾ 𝐵) “ {𝑧}) ≠ {𝑍}}) ↔ (𝑥 ∈ {𝑧 ∈ dom 𝐹 ∣ (𝐹 “ {𝑧}) ≠ {𝑍}} ∧ ¬ 𝑥 ∈ {𝑧 ∈ dom (𝐹 ↾ 𝐵) ∣ ((𝐹 ↾ 𝐵) “ {𝑧}) ≠ {𝑍}}))
2		sneq 4571	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → {𝑧} = {𝑥})
3	2	imaeq2d 5969	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝐹 “ {𝑧}) = (𝐹 “ {𝑥}))
4	3	neeq1d 3003	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → ((𝐹 “ {𝑧}) ≠ {𝑍} ↔ (𝐹 “ {𝑥}) ≠ {𝑍}))
5	4	elrab 3624	. . . . . . 7 ⊢ (𝑥 ∈ {𝑧 ∈ dom 𝐹 ∣ (𝐹 “ {𝑧}) ≠ {𝑍}} ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}))
6		ianor 979	. . . . . . . 8 ⊢ (¬ (𝑥 ∈ dom (𝐹 ↾ 𝐵) ∧ ((𝐹 ↾ 𝐵) “ {𝑥}) ≠ {𝑍}) ↔ (¬ 𝑥 ∈ dom (𝐹 ↾ 𝐵) ∨ ¬ ((𝐹 ↾ 𝐵) “ {𝑥}) ≠ {𝑍}))
7	2	imaeq2d 5969	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → ((𝐹 ↾ 𝐵) “ {𝑧}) = ((𝐹 ↾ 𝐵) “ {𝑥}))
8	7	neeq1d 3003	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (((𝐹 ↾ 𝐵) “ {𝑧}) ≠ {𝑍} ↔ ((𝐹 ↾ 𝐵) “ {𝑥}) ≠ {𝑍}))
9	8	elrab 3624	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑧 ∈ dom (𝐹 ↾ 𝐵) ∣ ((𝐹 ↾ 𝐵) “ {𝑧}) ≠ {𝑍}} ↔ (𝑥 ∈ dom (𝐹 ↾ 𝐵) ∧ ((𝐹 ↾ 𝐵) “ {𝑥}) ≠ {𝑍}))
10	6, 9	xchnxbir 333	. . . . . . 7 ⊢ (¬ 𝑥 ∈ {𝑧 ∈ dom (𝐹 ↾ 𝐵) ∣ ((𝐹 ↾ 𝐵) “ {𝑧}) ≠ {𝑍}} ↔ (¬ 𝑥 ∈ dom (𝐹 ↾ 𝐵) ∨ ¬ ((𝐹 ↾ 𝐵) “ {𝑥}) ≠ {𝑍}))
11		ianor 979	. . . . . . . . . . 11 ⊢ (¬ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ dom 𝐹) ↔ (¬ 𝑥 ∈ 𝐵 ∨ ¬ 𝑥 ∈ dom 𝐹))
12		dmres 5913	. . . . . . . . . . . 12 ⊢ dom (𝐹 ↾ 𝐵) = (𝐵 ∩ dom 𝐹)
13	12	elin2 4131	. . . . . . . . . . 11 ⊢ (𝑥 ∈ dom (𝐹 ↾ 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ dom 𝐹))
14	11, 13	xchnxbir 333	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ dom (𝐹 ↾ 𝐵) ↔ (¬ 𝑥 ∈ 𝐵 ∨ ¬ 𝑥 ∈ dom 𝐹))
15		simpl 483	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) → 𝑥 ∈ dom 𝐹)
16	15	anim2i 617	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝑥 ∈ 𝐵 ∧ (𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍})) → (¬ 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ dom 𝐹))
17	16	ancomd 462	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑥 ∈ 𝐵 ∧ (𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍})) → (𝑥 ∈ dom 𝐹 ∧ ¬ 𝑥 ∈ 𝐵))
18		eldif 3897	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (dom 𝐹 ∖ 𝐵) ↔ (𝑥 ∈ dom 𝐹 ∧ ¬ 𝑥 ∈ 𝐵))
19	17, 18	sylibr 233	. . . . . . . . . . . 12 ⊢ ((¬ 𝑥 ∈ 𝐵 ∧ (𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍})) → 𝑥 ∈ (dom 𝐹 ∖ 𝐵))
20	19	ex 413	. . . . . . . . . . 11 ⊢ (¬ 𝑥 ∈ 𝐵 → ((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) → 𝑥 ∈ (dom 𝐹 ∖ 𝐵)))
21		pm2.24 124	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ dom 𝐹 → (¬ 𝑥 ∈ dom 𝐹 → 𝑥 ∈ (dom 𝐹 ∖ 𝐵)))
22	21	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) → (¬ 𝑥 ∈ dom 𝐹 → 𝑥 ∈ (dom 𝐹 ∖ 𝐵)))
23	22	com12 32	. . . . . . . . . . 11 ⊢ (¬ 𝑥 ∈ dom 𝐹 → ((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) → 𝑥 ∈ (dom 𝐹 ∖ 𝐵)))
24	20, 23	jaoi 854	. . . . . . . . . 10 ⊢ ((¬ 𝑥 ∈ 𝐵 ∨ ¬ 𝑥 ∈ dom 𝐹) → ((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) → 𝑥 ∈ (dom 𝐹 ∖ 𝐵)))
25	14, 24	sylbi 216	. . . . . . . . 9 ⊢ (¬ 𝑥 ∈ dom (𝐹 ↾ 𝐵) → ((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) → 𝑥 ∈ (dom 𝐹 ∖ 𝐵)))
26	15	adantl 482	. . . . . . . . . . 11 ⊢ ((¬ ((𝐹 ↾ 𝐵) “ {𝑥}) ≠ {𝑍} ∧ (𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍})) → 𝑥 ∈ dom 𝐹)
27		snssi 4741	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ 𝐵 → {𝑥} ⊆ 𝐵)
28	27	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ 𝐵) → {𝑥} ⊆ 𝐵)
29		resima2 5926	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({𝑥} ⊆ 𝐵 → ((𝐹 ↾ 𝐵) “ {𝑥}) = (𝐹 “ {𝑥}))
30	28, 29	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ 𝐵) → ((𝐹 ↾ 𝐵) “ {𝑥}) = (𝐹 “ {𝑥}))
31	30	eqcomd 2744	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ 𝐵) → (𝐹 “ {𝑥}) = ((𝐹 ↾ 𝐵) “ {𝑥}))
32	31	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹 ↾ 𝐵) “ {𝑥}) = {𝑍}) → (𝐹 “ {𝑥}) = ((𝐹 ↾ 𝐵) “ {𝑥}))
33		simpr 485	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹 ↾ 𝐵) “ {𝑥}) = {𝑍}) → ((𝐹 ↾ 𝐵) “ {𝑥}) = {𝑍})
34	32, 33	eqtrd 2778	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹 ↾ 𝐵) “ {𝑥}) = {𝑍}) → (𝐹 “ {𝑥}) = {𝑍})
35	34	ex 413	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ 𝐵) → (((𝐹 ↾ 𝐵) “ {𝑥}) = {𝑍} → (𝐹 “ {𝑥}) = {𝑍}))
36	35	necon3d 2964	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ 𝐵) → ((𝐹 “ {𝑥}) ≠ {𝑍} → ((𝐹 ↾ 𝐵) “ {𝑥}) ≠ {𝑍}))
37	36	impancom 452	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) → (𝑥 ∈ 𝐵 → ((𝐹 ↾ 𝐵) “ {𝑥}) ≠ {𝑍}))
38	37	con3d 152	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) → (¬ ((𝐹 ↾ 𝐵) “ {𝑥}) ≠ {𝑍} → ¬ 𝑥 ∈ 𝐵))
39	38	impcom 408	. . . . . . . . . . 11 ⊢ ((¬ ((𝐹 ↾ 𝐵) “ {𝑥}) ≠ {𝑍} ∧ (𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍})) → ¬ 𝑥 ∈ 𝐵)
40	26, 39	eldifd 3898	. . . . . . . . . 10 ⊢ ((¬ ((𝐹 ↾ 𝐵) “ {𝑥}) ≠ {𝑍} ∧ (𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍})) → 𝑥 ∈ (dom 𝐹 ∖ 𝐵))
41	40	ex 413	. . . . . . . . 9 ⊢ (¬ ((𝐹 ↾ 𝐵) “ {𝑥}) ≠ {𝑍} → ((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) → 𝑥 ∈ (dom 𝐹 ∖ 𝐵)))
42	25, 41	jaoi 854	. . . . . . . 8 ⊢ ((¬ 𝑥 ∈ dom (𝐹 ↾ 𝐵) ∨ ¬ ((𝐹 ↾ 𝐵) “ {𝑥}) ≠ {𝑍}) → ((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) → 𝑥 ∈ (dom 𝐹 ∖ 𝐵)))
43	42	impcom 408	. . . . . . 7 ⊢ (((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) ∧ (¬ 𝑥 ∈ dom (𝐹 ↾ 𝐵) ∨ ¬ ((𝐹 ↾ 𝐵) “ {𝑥}) ≠ {𝑍})) → 𝑥 ∈ (dom 𝐹 ∖ 𝐵))
44	5, 10, 43	syl2anb 598	. . . . . 6 ⊢ ((𝑥 ∈ {𝑧 ∈ dom 𝐹 ∣ (𝐹 “ {𝑧}) ≠ {𝑍}} ∧ ¬ 𝑥 ∈ {𝑧 ∈ dom (𝐹 ↾ 𝐵) ∣ ((𝐹 ↾ 𝐵) “ {𝑧}) ≠ {𝑍}}) → 𝑥 ∈ (dom 𝐹 ∖ 𝐵))
45	1, 44	sylbi 216	. . . . 5 ⊢ (𝑥 ∈ ({𝑧 ∈ dom 𝐹 ∣ (𝐹 “ {𝑧}) ≠ {𝑍}} ∖ {𝑧 ∈ dom (𝐹 ↾ 𝐵) ∣ ((𝐹 ↾ 𝐵) “ {𝑧}) ≠ {𝑍}}) → 𝑥 ∈ (dom 𝐹 ∖ 𝐵))
46	45	a1i 11	. . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑥 ∈ ({𝑧 ∈ dom 𝐹 ∣ (𝐹 “ {𝑧}) ≠ {𝑍}} ∖ {𝑧 ∈ dom (𝐹 ↾ 𝐵) ∣ ((𝐹 ↾ 𝐵) “ {𝑧}) ≠ {𝑍}}) → 𝑥 ∈ (dom 𝐹 ∖ 𝐵)))
47	46	ssrdv 3927	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ({𝑧 ∈ dom 𝐹 ∣ (𝐹 “ {𝑧}) ≠ {𝑍}} ∖ {𝑧 ∈ dom (𝐹 ↾ 𝐵) ∣ ((𝐹 ↾ 𝐵) “ {𝑧}) ≠ {𝑍}}) ⊆ (dom 𝐹 ∖ 𝐵))
48		ssundif 4418	. . 3 ⊢ ({𝑧 ∈ dom 𝐹 ∣ (𝐹 “ {𝑧}) ≠ {𝑍}} ⊆ ({𝑧 ∈ dom (𝐹 ↾ 𝐵) ∣ ((𝐹 ↾ 𝐵) “ {𝑧}) ≠ {𝑍}} ∪ (dom 𝐹 ∖ 𝐵)) ↔ ({𝑧 ∈ dom 𝐹 ∣ (𝐹 “ {𝑧}) ≠ {𝑍}} ∖ {𝑧 ∈ dom (𝐹 ↾ 𝐵) ∣ ((𝐹 ↾ 𝐵) “ {𝑧}) ≠ {𝑍}}) ⊆ (dom 𝐹 ∖ 𝐵))
49	47, 48	sylibr 233	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → {𝑧 ∈ dom 𝐹 ∣ (𝐹 “ {𝑧}) ≠ {𝑍}} ⊆ ({𝑧 ∈ dom (𝐹 ↾ 𝐵) ∣ ((𝐹 ↾ 𝐵) “ {𝑧}) ≠ {𝑍}} ∪ (dom 𝐹 ∖ 𝐵)))
50		suppval 7979	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = {𝑧 ∈ dom 𝐹 ∣ (𝐹 “ {𝑧}) ≠ {𝑍}})
51		resexg 5937	. . . 4 ⊢ (𝐹 ∈ 𝑉 → (𝐹 ↾ 𝐵) ∈ V)
52		suppval 7979	. . . 4 ⊢ (((𝐹 ↾ 𝐵) ∈ V ∧ 𝑍 ∈ 𝑊) → ((𝐹 ↾ 𝐵) supp 𝑍) = {𝑧 ∈ dom (𝐹 ↾ 𝐵) ∣ ((𝐹 ↾ 𝐵) “ {𝑧}) ≠ {𝑍}})
53	51, 52	sylan 580	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ((𝐹 ↾ 𝐵) supp 𝑍) = {𝑧 ∈ dom (𝐹 ↾ 𝐵) ∣ ((𝐹 ↾ 𝐵) “ {𝑧}) ≠ {𝑍}})
54	53	uneq1d 4096	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (((𝐹 ↾ 𝐵) supp 𝑍) ∪ (dom 𝐹 ∖ 𝐵)) = ({𝑧 ∈ dom (𝐹 ↾ 𝐵) ∣ ((𝐹 ↾ 𝐵) “ {𝑧}) ≠ {𝑍}} ∪ (dom 𝐹 ∖ 𝐵)))
55	49, 50, 54	3sstr4d 3968	1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) ⊆ (((𝐹 ↾ 𝐵) supp 𝑍) ∪ (dom 𝐹 ∖ 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∨ wo 844   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  {crab 3068  Vcvv 3432   ∖ cdif 3884   ∪ cun 3885   ⊆ wss 3887  {csn 4561  dom cdm 5589   ↾ cres 5591   “ cima 5592  (class class class)co 7275   supp csupp 7977

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-supp 7978

This theorem is referenced by:  ressuppfi  9154