Theorem suppssov1 8194

Description: Formula building theorem for support restrictions: operator with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 28-May-2019.) (Proof shortened by SN, 11-Apr-2025.)

Hypotheses

Ref	Expression
suppssov1.s	⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ 𝐴) supp 𝑌) ⊆ 𝐿)
suppssov1.o	⊢ ((𝜑 ∧ 𝑣 ∈ 𝑅) → (𝑌𝑂𝑣) = 𝑍)
suppssov1.a	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐴 ∈ 𝑉)
suppssov1.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐵 ∈ 𝑅)
suppssov1.y	⊢ (𝜑 → 𝑌 ∈ 𝑊)

Assertion

Ref	Expression
suppssov1	⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) ⊆ 𝐿)

Distinct variable groups:   𝜑,𝑣   𝜑,𝑥   𝑣,𝐵   𝑥,𝐷   𝑣,𝑂   𝑣,𝑅   𝑣,𝑌   𝑥,𝑌   𝑣,𝑍   𝑥,𝑍

Allowed substitution hints:   𝐴(𝑥,𝑣)   𝐵(𝑥)   𝐷(𝑣)   𝑅(𝑥)   𝐿(𝑥,𝑣)   𝑂(𝑥)   𝑉(𝑥,𝑣)   𝑊(𝑥,𝑣)

Proof of Theorem suppssov1

Step	Hyp	Ref	Expression
1		suppssov1.a	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐴 ∈ 𝑉)
2	1	elexd 3478	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐴 ∈ V)
3	2	adantlr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥 ∈ 𝐷) → 𝐴 ∈ V)
4	3	adantr 485	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})) → 𝐴 ∈ V)
5		oveq2 7420	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝐵 → (𝑌𝑂𝑣) = (𝑌𝑂𝐵))
6	5	eqeq1d 2765	. . . . . . . . . . 11 ⊢ (𝑣 = 𝐵 → ((𝑌𝑂𝑣) = 𝑍 ↔ (𝑌𝑂𝐵) = 𝑍))
7		suppssov1.o	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑅) → (𝑌𝑂𝑣) = 𝑍)
8	7	ralrimiva 3157	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑣 ∈ 𝑅 (𝑌𝑂𝑣) = 𝑍)
9	8	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥 ∈ 𝐷) → ∀𝑣 ∈ 𝑅 (𝑌𝑂𝑣) = 𝑍)
10		suppssov1.b	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐵 ∈ 𝑅)
11	10	adantlr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥 ∈ 𝐷) → 𝐵 ∈ 𝑅)
12	6, 9, 11	rspcdva 3583	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥 ∈ 𝐷) → (𝑌𝑂𝐵) = 𝑍)
13		oveq1 7419	. . . . . . . . . . 11 ⊢ (𝐴 = 𝑌 → (𝐴𝑂𝐵) = (𝑌𝑂𝐵))
14	13	eqeq1d 2765	. . . . . . . . . 10 ⊢ (𝐴 = 𝑌 → ((𝐴𝑂𝐵) = 𝑍 ↔ (𝑌𝑂𝐵) = 𝑍))
15	12, 14	syl5ibrcom 250	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥 ∈ 𝐷) → (𝐴 = 𝑌 → (𝐴𝑂𝐵) = 𝑍))
16	15	necon3d 2979	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥 ∈ 𝐷) → ((𝐴𝑂𝐵) ≠ 𝑍 → 𝐴 ≠ 𝑌))
17		eldifsni 4759	. . . . . . . 8 ⊢ ((𝐴𝑂𝐵) ∈ (V ∖ {𝑍}) → (𝐴𝑂𝐵) ≠ 𝑍)
18	16, 17	impel 514	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})) → 𝐴 ≠ 𝑌)
19		eldifsn 4754	. . . . . . 7 ⊢ (𝐴 ∈ (V ∖ {𝑌}) ↔ (𝐴 ∈ V ∧ 𝐴 ≠ 𝑌))
20	4, 18, 19	sylanbrc 594	. . . . . 6 ⊢ ((((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})) → 𝐴 ∈ (V ∖ {𝑌}))
21	20	ex 417	. . . . 5 ⊢ (((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥 ∈ 𝐷) → ((𝐴𝑂𝐵) ∈ (V ∖ {𝑍}) → 𝐴 ∈ (V ∖ {𝑌})))
22	21	ss2rabdv 4030	. . . 4 ⊢ ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → {𝑥 ∈ 𝐷 ∣ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})} ⊆ {𝑥 ∈ 𝐷 ∣ 𝐴 ∈ (V ∖ {𝑌})})
23		eqid 2763	. . . . 5 ⊢ (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) = (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵))
24		simprl 782	. . . . 5 ⊢ ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → 𝐷 ∈ V)
25		simprr 784	. . . . 5 ⊢ ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → 𝑍 ∈ V)
26	23, 24, 25	mptsuppdifd 8183	. . . 4 ⊢ ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → ((𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) = {𝑥 ∈ 𝐷 ∣ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})})
27		eqid 2763	. . . . 5 ⊢ (𝑥 ∈ 𝐷 ↦ 𝐴) = (𝑥 ∈ 𝐷 ↦ 𝐴)
28		suppssov1.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑊)
29	28	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → 𝑌 ∈ 𝑊)
30	27, 24, 29	mptsuppdifd 8183	. . . 4 ⊢ ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → ((𝑥 ∈ 𝐷 ↦ 𝐴) supp 𝑌) = {𝑥 ∈ 𝐷 ∣ 𝐴 ∈ (V ∖ {𝑌})})
31	22, 26, 30	3sstr4d 3993	. . 3 ⊢ ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → ((𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) ⊆ ((𝑥 ∈ 𝐷 ↦ 𝐴) supp 𝑌))
32		suppssov1.s	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ 𝐴) supp 𝑌) ⊆ 𝐿)
33	32	adantr 485	. . 3 ⊢ ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → ((𝑥 ∈ 𝐷 ↦ 𝐴) supp 𝑌) ⊆ 𝐿)
34	31, 33	sstrd 3948	. 2 ⊢ ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → ((𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) ⊆ 𝐿)
35		mptexg 7221	. . . . . . 7 ⊢ (𝐷 ∈ V → (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) ∈ V)
36		ovex 7445	. . . . . . . . . 10 ⊢ (𝐴𝑂𝐵) ∈ V
37	36	rgenw 3083	. . . . . . . . 9 ⊢ ∀𝑥 ∈ 𝐷 (𝐴𝑂𝐵) ∈ V
38		dmmptg 6245	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐷 (𝐴𝑂𝐵) ∈ V → dom (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) = 𝐷)
39	37, 38	ax-mp 5	. . . . . . . 8 ⊢ dom (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) = 𝐷
40		dmexg 7899	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) ∈ V → dom (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) ∈ V)
41	39, 40	eqeltrrid 2868	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) ∈ V → 𝐷 ∈ V)
42	35, 41	impbii 212	. . . . . 6 ⊢ (𝐷 ∈ V ↔ (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) ∈ V)
43	42	anbi1i 635	. . . . 5 ⊢ ((𝐷 ∈ V ∧ 𝑍 ∈ V) ↔ ((𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) ∈ V ∧ 𝑍 ∈ V))
44		supp0prc 8160	. . . . 5 ⊢ (¬ ((𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) ∈ V ∧ 𝑍 ∈ V) → ((𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) = ∅)
45	43, 44	sylnbi 333	. . . 4 ⊢ (¬ (𝐷 ∈ V ∧ 𝑍 ∈ V) → ((𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) = ∅)
46		0ss 4358	. . . 4 ⊢ ∅ ⊆ 𝐿
47	45, 46	eqsstrdi 3982	. . 3 ⊢ (¬ (𝐷 ∈ V ∧ 𝑍 ∈ V) → ((𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) ⊆ 𝐿)
48	47	adantl 486	. 2 ⊢ ((𝜑 ∧ ¬ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → ((𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) ⊆ 𝐿)
49	34, 48	pm2.61dan 824	1 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) ⊆ 𝐿)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  {crab 3416  Vcvv 3455   ∖ cdif 3903   ⊆ wss 3906  ∅c0 4287  {csn 4590   ↦ cmpt 5193  dom cdm 5663  (class class class)co 7412   supp csupp 8157

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-supp 8158

This theorem is referenced by:  suppssof1  8196  fsuppssov1  9345  evlslem6  22213  plypf1  26350  fisuppov1  33006  mhphf  43309