Theorem suppssov2 8194

Description: Formula building theorem for support restrictions: operator with right annihilator. (Contributed by SN, 11-Apr-2025.)

Hypotheses

Ref	Expression
suppssov2.s	⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ 𝐵) supp 𝑌) ⊆ 𝐿)
suppssov2.o	⊢ ((𝜑 ∧ 𝑣 ∈ 𝑅) → (𝑣𝑂𝑌) = 𝑍)
suppssov2.a	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐴 ∈ 𝑅)
suppssov2.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐵 ∈ 𝑉)
suppssov2.y	⊢ (𝜑 → 𝑌 ∈ 𝑊)

Assertion

Ref	Expression
suppssov2	⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) ⊆ 𝐿)

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

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

Proof of Theorem suppssov2

Step	Hyp	Ref	Expression
1		suppssov2.b	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐵 ∈ 𝑉)
2	1	elexd 3486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐵 ∈ V)
3	2	adantlr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥 ∈ 𝐷) → 𝐵 ∈ V)
4	3	adantr 485	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})) → 𝐵 ∈ V)
5		oveq1 7418	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝐴 → (𝑣𝑂𝑌) = (𝐴𝑂𝑌))
6	5	eqeq1d 2771	. . . . . . . . . . 11 ⊢ (𝑣 = 𝐴 → ((𝑣𝑂𝑌) = 𝑍 ↔ (𝐴𝑂𝑌) = 𝑍))
7		suppssov2.o	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑅) → (𝑣𝑂𝑌) = 𝑍)
8	7	ralrimiva 3163	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑣 ∈ 𝑅 (𝑣𝑂𝑌) = 𝑍)
9	8	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥 ∈ 𝐷) → ∀𝑣 ∈ 𝑅 (𝑣𝑂𝑌) = 𝑍)
10		suppssov2.a	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐴 ∈ 𝑅)
11	10	adantlr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥 ∈ 𝐷) → 𝐴 ∈ 𝑅)
12	6, 9, 11	rspcdva 3591	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥 ∈ 𝐷) → (𝐴𝑂𝑌) = 𝑍)
13		oveq2 7419	. . . . . . . . . . 11 ⊢ (𝐵 = 𝑌 → (𝐴𝑂𝐵) = (𝐴𝑂𝑌))
14	13	eqeq1d 2771	. . . . . . . . . 10 ⊢ (𝐵 = 𝑌 → ((𝐴𝑂𝐵) = 𝑍 ↔ (𝐴𝑂𝑌) = 𝑍))
15	12, 14	syl5ibrcom 250	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥 ∈ 𝐷) → (𝐵 = 𝑌 → (𝐴𝑂𝐵) = 𝑍))
16	15	necon3d 2985	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥 ∈ 𝐷) → ((𝐴𝑂𝐵) ≠ 𝑍 → 𝐵 ≠ 𝑌))
17		eldifsni 4762	. . . . . . . 8 ⊢ ((𝐴𝑂𝐵) ∈ (V ∖ {𝑍}) → (𝐴𝑂𝐵) ≠ 𝑍)
18	16, 17	impel 514	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})) → 𝐵 ≠ 𝑌)
19		eldifsn 4758	. . . . . . 7 ⊢ (𝐵 ∈ (V ∖ {𝑌}) ↔ (𝐵 ∈ V ∧ 𝐵 ≠ 𝑌))
20	4, 18, 19	sylanbrc 594	. . . . . 6 ⊢ ((((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})) → 𝐵 ∈ (V ∖ {𝑌}))
21	20	ex 417	. . . . 5 ⊢ (((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥 ∈ 𝐷) → ((𝐴𝑂𝐵) ∈ (V ∖ {𝑍}) → 𝐵 ∈ (V ∖ {𝑌})))
22	21	ss2rabdv 4037	. . . 4 ⊢ ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → {𝑥 ∈ 𝐷 ∣ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})} ⊆ {𝑥 ∈ 𝐷 ∣ 𝐵 ∈ (V ∖ {𝑌})})
23		eqid 2769	. . . . 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 8182	. . . 4 ⊢ ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → ((𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) = {𝑥 ∈ 𝐷 ∣ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})})
27		eqid 2769	. . . . 5 ⊢ (𝑥 ∈ 𝐷 ↦ 𝐵) = (𝑥 ∈ 𝐷 ↦ 𝐵)
28		suppssov2.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑊)
29	28	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → 𝑌 ∈ 𝑊)
30	27, 24, 29	mptsuppdifd 8182	. . . 4 ⊢ ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → ((𝑥 ∈ 𝐷 ↦ 𝐵) supp 𝑌) = {𝑥 ∈ 𝐷 ∣ 𝐵 ∈ (V ∖ {𝑌})})
31	22, 26, 30	3sstr4d 4000	. . 3 ⊢ ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → ((𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) ⊆ ((𝑥 ∈ 𝐷 ↦ 𝐵) supp 𝑌))
32		suppssov2.s	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ 𝐵) supp 𝑌) ⊆ 𝐿)
33	32	adantr 485	. . 3 ⊢ ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → ((𝑥 ∈ 𝐷 ↦ 𝐵) supp 𝑌) ⊆ 𝐿)
34	31, 33	sstrd 3955	. 2 ⊢ ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → ((𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) ⊆ 𝐿)
35		mptexg 7220	. . . . . . 7 ⊢ (𝐷 ∈ V → (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) ∈ V)
36		ovex 7444	. . . . . . . . . 10 ⊢ (𝐴𝑂𝐵) ∈ V
37	36	rgenw 3089	. . . . . . . . 9 ⊢ ∀𝑥 ∈ 𝐷 (𝐴𝑂𝐵) ∈ V
38		dmmptg 6244	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐷 (𝐴𝑂𝐵) ∈ V → dom (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) = 𝐷)
39	37, 38	ax-mp 5	. . . . . . . 8 ⊢ dom (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) = 𝐷
40		dmexg 7898	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) ∈ V → dom (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) ∈ V)
41	39, 40	eqeltrrid 2874	. . . . . . 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 8159	. . . . 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 4364	. . . 4 ⊢ ∅ ⊆ 𝐿
47	45, 46	eqsstrdi 3989	. . 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 1567   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  {crab 3423  Vcvv 3463   ∖ cdif 3910   ⊆ wss 3913  ∅c0 4294  {csn 4594   ↦ cmpt 5196  dom cdm 5662  (class class class)co 7411   supp csupp 8156

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-supp 8157

This theorem is referenced by:  psdmplcl  22294