Theorem suppssov1 8238

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 3512	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐴 ∈ V)
3	2	adantlr 714	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥 ∈ 𝐷) → 𝐴 ∈ V)
4	3	adantr 480	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})) → 𝐴 ∈ V)
5		oveq2 7456	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝐵 → (𝑌𝑂𝑣) = (𝑌𝑂𝐵))
6	5	eqeq1d 2742	. . . . . . . . . . 11 ⊢ (𝑣 = 𝐵 → ((𝑌𝑂𝑣) = 𝑍 ↔ (𝑌𝑂𝐵) = 𝑍))
7		suppssov1.o	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑅) → (𝑌𝑂𝑣) = 𝑍)
8	7	ralrimiva 3152	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑣 ∈ 𝑅 (𝑌𝑂𝑣) = 𝑍)
9	8	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥 ∈ 𝐷) → ∀𝑣 ∈ 𝑅 (𝑌𝑂𝑣) = 𝑍)
10		suppssov1.b	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐵 ∈ 𝑅)
11	10	adantlr 714	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥 ∈ 𝐷) → 𝐵 ∈ 𝑅)
12	6, 9, 11	rspcdva 3636	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥 ∈ 𝐷) → (𝑌𝑂𝐵) = 𝑍)
13		oveq1 7455	. . . . . . . . . . 11 ⊢ (𝐴 = 𝑌 → (𝐴𝑂𝐵) = (𝑌𝑂𝐵))
14	13	eqeq1d 2742	. . . . . . . . . 10 ⊢ (𝐴 = 𝑌 → ((𝐴𝑂𝐵) = 𝑍 ↔ (𝑌𝑂𝐵) = 𝑍))
15	12, 14	syl5ibrcom 247	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥 ∈ 𝐷) → (𝐴 = 𝑌 → (𝐴𝑂𝐵) = 𝑍))
16	15	necon3d 2967	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥 ∈ 𝐷) → ((𝐴𝑂𝐵) ≠ 𝑍 → 𝐴 ≠ 𝑌))
17		eldifsni 4815	. . . . . . . 8 ⊢ ((𝐴𝑂𝐵) ∈ (V ∖ {𝑍}) → (𝐴𝑂𝐵) ≠ 𝑍)
18	16, 17	impel 505	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})) → 𝐴 ≠ 𝑌)
19		eldifsn 4811	. . . . . . 7 ⊢ (𝐴 ∈ (V ∖ {𝑌}) ↔ (𝐴 ∈ V ∧ 𝐴 ≠ 𝑌))
20	4, 18, 19	sylanbrc 582	. . . . . 6 ⊢ ((((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})) → 𝐴 ∈ (V ∖ {𝑌}))
21	20	ex 412	. . . . 5 ⊢ (((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥 ∈ 𝐷) → ((𝐴𝑂𝐵) ∈ (V ∖ {𝑍}) → 𝐴 ∈ (V ∖ {𝑌})))
22	21	ss2rabdv 4099	. . . 4 ⊢ ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → {𝑥 ∈ 𝐷 ∣ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})} ⊆ {𝑥 ∈ 𝐷 ∣ 𝐴 ∈ (V ∖ {𝑌})})
23		eqid 2740	. . . . 5 ⊢ (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) = (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵))
24		simprl 770	. . . . 5 ⊢ ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → 𝐷 ∈ V)
25		simprr 772	. . . . 5 ⊢ ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → 𝑍 ∈ V)
26	23, 24, 25	mptsuppdifd 8227	. . . 4 ⊢ ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → ((𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) = {𝑥 ∈ 𝐷 ∣ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})})
27		eqid 2740	. . . . 5 ⊢ (𝑥 ∈ 𝐷 ↦ 𝐴) = (𝑥 ∈ 𝐷 ↦ 𝐴)
28		suppssov1.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑊)
29	28	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → 𝑌 ∈ 𝑊)
30	27, 24, 29	mptsuppdifd 8227	. . . 4 ⊢ ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → ((𝑥 ∈ 𝐷 ↦ 𝐴) supp 𝑌) = {𝑥 ∈ 𝐷 ∣ 𝐴 ∈ (V ∖ {𝑌})})
31	22, 26, 30	3sstr4d 4056	. . 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 480	. . 3 ⊢ ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → ((𝑥 ∈ 𝐷 ↦ 𝐴) supp 𝑌) ⊆ 𝐿)
34	31, 33	sstrd 4019	. 2 ⊢ ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → ((𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) ⊆ 𝐿)
35		mptexg 7258	. . . . . . 7 ⊢ (𝐷 ∈ V → (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) ∈ V)
36		ovex 7481	. . . . . . . . . 10 ⊢ (𝐴𝑂𝐵) ∈ V
37	36	rgenw 3071	. . . . . . . . 9 ⊢ ∀𝑥 ∈ 𝐷 (𝐴𝑂𝐵) ∈ V
38		dmmptg 6273	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐷 (𝐴𝑂𝐵) ∈ V → dom (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) = 𝐷)
39	37, 38	ax-mp 5	. . . . . . . 8 ⊢ dom (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) = 𝐷
40		dmexg 7941	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) ∈ V → dom (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) ∈ V)
41	39, 40	eqeltrrid 2849	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) ∈ V → 𝐷 ∈ V)
42	35, 41	impbii 209	. . . . . 6 ⊢ (𝐷 ∈ V ↔ (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) ∈ V)
43	42	anbi1i 623	. . . . 5 ⊢ ((𝐷 ∈ V ∧ 𝑍 ∈ V) ↔ ((𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) ∈ V ∧ 𝑍 ∈ V))
44		supp0prc 8204	. . . . 5 ⊢ (¬ ((𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) ∈ V ∧ 𝑍 ∈ V) → ((𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) = ∅)
45	43, 44	sylnbi 330	. . . 4 ⊢ (¬ (𝐷 ∈ V ∧ 𝑍 ∈ V) → ((𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) = ∅)
46		0ss 4423	. . . 4 ⊢ ∅ ⊆ 𝐿
47	45, 46	eqsstrdi 4063	. . 3 ⊢ (¬ (𝐷 ∈ V ∧ 𝑍 ∈ V) → ((𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) ⊆ 𝐿)
48	47	adantl 481	. 2 ⊢ ((𝜑 ∧ ¬ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → ((𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) ⊆ 𝐿)
49	34, 48	pm2.61dan 812	1 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) ⊆ 𝐿)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  {crab 3443  Vcvv 3488   ∖ cdif 3973   ⊆ wss 3976  ∅c0 4352  {csn 4648   ↦ cmpt 5249  dom cdm 5700  (class class class)co 7448   supp csupp 8201

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-supp 8202

This theorem is referenced by:  suppssof1  8240  fsuppssov1  9453  evlslem6  22128  plypf1  26271  mhphf  42552