Theorem ressuppssdif 7580

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 3808	. . . . . 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 4407	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → {𝑧} = {𝑥})
3	2	imaeq2d 5707	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝐹 “ {𝑧}) = (𝐹 “ {𝑥}))
4	3	neeq1d 3058	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → ((𝐹 “ {𝑧}) ≠ {𝑍} ↔ (𝐹 “ {𝑥}) ≠ {𝑍}))
5	4	elrab 3585	. . . . . . 7 ⊢ (𝑥 ∈ {𝑧 ∈ dom 𝐹 ∣ (𝐹 “ {𝑧}) ≠ {𝑍}} ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}))
6		ianor 1011	. . . . . . . 8 ⊢ (¬ (𝑥 ∈ dom (𝐹 ↾ 𝐵) ∧ ((𝐹 ↾ 𝐵) “ {𝑥}) ≠ {𝑍}) ↔ (¬ 𝑥 ∈ dom (𝐹 ↾ 𝐵) ∨ ¬ ((𝐹 ↾ 𝐵) “ {𝑥}) ≠ {𝑍}))
7	2	imaeq2d 5707	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → ((𝐹 ↾ 𝐵) “ {𝑧}) = ((𝐹 ↾ 𝐵) “ {𝑥}))
8	7	neeq1d 3058	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (((𝐹 ↾ 𝐵) “ {𝑧}) ≠ {𝑍} ↔ ((𝐹 ↾ 𝐵) “ {𝑥}) ≠ {𝑍}))
9	8	elrab 3585	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑧 ∈ dom (𝐹 ↾ 𝐵) ∣ ((𝐹 ↾ 𝐵) “ {𝑧}) ≠ {𝑍}} ↔ (𝑥 ∈ dom (𝐹 ↾ 𝐵) ∧ ((𝐹 ↾ 𝐵) “ {𝑥}) ≠ {𝑍}))
10	6, 9	xchnxbir 325	. . . . . . 7 ⊢ (¬ 𝑥 ∈ {𝑧 ∈ dom (𝐹 ↾ 𝐵) ∣ ((𝐹 ↾ 𝐵) “ {𝑧}) ≠ {𝑍}} ↔ (¬ 𝑥 ∈ dom (𝐹 ↾ 𝐵) ∨ ¬ ((𝐹 ↾ 𝐵) “ {𝑥}) ≠ {𝑍}))
11		ianor 1011	. . . . . . . . . . 11 ⊢ (¬ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ dom 𝐹) ↔ (¬ 𝑥 ∈ 𝐵 ∨ ¬ 𝑥 ∈ dom 𝐹))
12		dmres 5655	. . . . . . . . . . . 12 ⊢ dom (𝐹 ↾ 𝐵) = (𝐵 ∩ dom 𝐹)
13	12	elin2 4028	. . . . . . . . . . 11 ⊢ (𝑥 ∈ dom (𝐹 ↾ 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ dom 𝐹))
14	11, 13	xchnxbir 325	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ dom (𝐹 ↾ 𝐵) ↔ (¬ 𝑥 ∈ 𝐵 ∨ ¬ 𝑥 ∈ dom 𝐹))
15		simpl 476	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) → 𝑥 ∈ dom 𝐹)
16	15	anim2i 612	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝑥 ∈ 𝐵 ∧ (𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍})) → (¬ 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ dom 𝐹))
17	16	ancomd 455	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑥 ∈ 𝐵 ∧ (𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍})) → (𝑥 ∈ dom 𝐹 ∧ ¬ 𝑥 ∈ 𝐵))
18		eldif 3808	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (dom 𝐹 ∖ 𝐵) ↔ (𝑥 ∈ dom 𝐹 ∧ ¬ 𝑥 ∈ 𝐵))
19	17, 18	sylibr 226	. . . . . . . . . . . 12 ⊢ ((¬ 𝑥 ∈ 𝐵 ∧ (𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍})) → 𝑥 ∈ (dom 𝐹 ∖ 𝐵))
20	19	ex 403	. . . . . . . . . . 11 ⊢ (¬ 𝑥 ∈ 𝐵 → ((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) → 𝑥 ∈ (dom 𝐹 ∖ 𝐵)))
21		pm2.24 122	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ dom 𝐹 → (¬ 𝑥 ∈ dom 𝐹 → 𝑥 ∈ (dom 𝐹 ∖ 𝐵)))
22	21	adantr 474	. . . . . . . . . . . 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 890	. . . . . . . . . 10 ⊢ ((¬ 𝑥 ∈ 𝐵 ∨ ¬ 𝑥 ∈ dom 𝐹) → ((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) → 𝑥 ∈ (dom 𝐹 ∖ 𝐵)))
25	14, 24	sylbi 209	. . . . . . . . 9 ⊢ (¬ 𝑥 ∈ dom (𝐹 ↾ 𝐵) → ((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) → 𝑥 ∈ (dom 𝐹 ∖ 𝐵)))
26	15	adantl 475	. . . . . . . . . . 11 ⊢ ((¬ ((𝐹 ↾ 𝐵) “ {𝑥}) ≠ {𝑍} ∧ (𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍})) → 𝑥 ∈ dom 𝐹)
27		snssi 4557	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ 𝐵 → {𝑥} ⊆ 𝐵)
28	27	adantl 475	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ 𝐵) → {𝑥} ⊆ 𝐵)
29		resima2 5668	. . . . . . . . . . . . . . . . . . . 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 2831	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ 𝐵) → (𝐹 “ {𝑥}) = ((𝐹 ↾ 𝐵) “ {𝑥}))
32	31	adantr 474	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹 ↾ 𝐵) “ {𝑥}) = {𝑍}) → (𝐹 “ {𝑥}) = ((𝐹 ↾ 𝐵) “ {𝑥}))
33		simpr 479	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹 ↾ 𝐵) “ {𝑥}) = {𝑍}) → ((𝐹 ↾ 𝐵) “ {𝑥}) = {𝑍})
34	32, 33	eqtrd 2861	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹 ↾ 𝐵) “ {𝑥}) = {𝑍}) → (𝐹 “ {𝑥}) = {𝑍})
35	34	ex 403	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ 𝐵) → (((𝐹 ↾ 𝐵) “ {𝑥}) = {𝑍} → (𝐹 “ {𝑥}) = {𝑍}))
36	35	necon3d 3020	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ 𝐵) → ((𝐹 “ {𝑥}) ≠ {𝑍} → ((𝐹 ↾ 𝐵) “ {𝑥}) ≠ {𝑍}))
37	36	impancom 445	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) → (𝑥 ∈ 𝐵 → ((𝐹 ↾ 𝐵) “ {𝑥}) ≠ {𝑍}))
38	37	con3d 150	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) → (¬ ((𝐹 ↾ 𝐵) “ {𝑥}) ≠ {𝑍} → ¬ 𝑥 ∈ 𝐵))
39	38	impcom 398	. . . . . . . . . . 11 ⊢ ((¬ ((𝐹 ↾ 𝐵) “ {𝑥}) ≠ {𝑍} ∧ (𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍})) → ¬ 𝑥 ∈ 𝐵)
40	26, 39	eldifd 3809	. . . . . . . . . 10 ⊢ ((¬ ((𝐹 ↾ 𝐵) “ {𝑥}) ≠ {𝑍} ∧ (𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍})) → 𝑥 ∈ (dom 𝐹 ∖ 𝐵))
41	40	ex 403	. . . . . . . . 9 ⊢ (¬ ((𝐹 ↾ 𝐵) “ {𝑥}) ≠ {𝑍} → ((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) → 𝑥 ∈ (dom 𝐹 ∖ 𝐵)))
42	25, 41	jaoi 890	. . . . . . . 8 ⊢ ((¬ 𝑥 ∈ dom (𝐹 ↾ 𝐵) ∨ ¬ ((𝐹 ↾ 𝐵) “ {𝑥}) ≠ {𝑍}) → ((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) → 𝑥 ∈ (dom 𝐹 ∖ 𝐵)))
43	42	impcom 398	. . . . . . 7 ⊢ (((𝑥 ∈ dom 𝐹 ∧ (𝐹 “ {𝑥}) ≠ {𝑍}) ∧ (¬ 𝑥 ∈ dom (𝐹 ↾ 𝐵) ∨ ¬ ((𝐹 ↾ 𝐵) “ {𝑥}) ≠ {𝑍})) → 𝑥 ∈ (dom 𝐹 ∖ 𝐵))
44	5, 10, 43	syl2anb 593	. . . . . 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 209	. . . . 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 3833	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ({𝑧 ∈ dom 𝐹 ∣ (𝐹 “ {𝑧}) ≠ {𝑍}} ∖ {𝑧 ∈ dom (𝐹 ↾ 𝐵) ∣ ((𝐹 ↾ 𝐵) “ {𝑧}) ≠ {𝑍}}) ⊆ (dom 𝐹 ∖ 𝐵))
48		ssundif 4275	. . 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 226	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → {𝑧 ∈ dom 𝐹 ∣ (𝐹 “ {𝑧}) ≠ {𝑍}} ⊆ ({𝑧 ∈ dom (𝐹 ↾ 𝐵) ∣ ((𝐹 ↾ 𝐵) “ {𝑧}) ≠ {𝑍}} ∪ (dom 𝐹 ∖ 𝐵)))
50		suppval 7561	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = {𝑧 ∈ dom 𝐹 ∣ (𝐹 “ {𝑧}) ≠ {𝑍}})
51		resexg 5679	. . . 4 ⊢ (𝐹 ∈ 𝑉 → (𝐹 ↾ 𝐵) ∈ V)
52		suppval 7561	. . . 4 ⊢ (((𝐹 ↾ 𝐵) ∈ V ∧ 𝑍 ∈ 𝑊) → ((𝐹 ↾ 𝐵) supp 𝑍) = {𝑧 ∈ dom (𝐹 ↾ 𝐵) ∣ ((𝐹 ↾ 𝐵) “ {𝑧}) ≠ {𝑍}})
53	51, 52	sylan 577	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ((𝐹 ↾ 𝐵) supp 𝑍) = {𝑧 ∈ dom (𝐹 ↾ 𝐵) ∣ ((𝐹 ↾ 𝐵) “ {𝑧}) ≠ {𝑍}})
54	53	uneq1d 3993	. 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 3873	1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) ⊆ (((𝐹 ↾ 𝐵) supp 𝑍) ∪ (dom 𝐹 ∖ 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 386   ∨ wo 880   = wceq 1658   ∈ wcel 2166   ≠ wne 2999  {crab 3121  Vcvv 3414   ∖ cdif 3795   ∪ cun 3796   ⊆ wss 3798  {csn 4397  dom cdm 5342   ↾ cres 5344   “ cima 5345  (class class class)co 6905   supp csupp 7559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127  ax-un 7209

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-supp 7560

This theorem is referenced by:  ressuppfi  8570