Theorem suppssov2 8179

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 3474	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐵 ∈ V)
3	2	adantlr 715	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥 ∈ 𝐷) → 𝐵 ∈ V)
4	3	adantr 480	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})) → 𝐵 ∈ V)
5		oveq1 7396	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝐴 → (𝑣𝑂𝑌) = (𝐴𝑂𝑌))
6	5	eqeq1d 2732	. . . . . . . . . . 11 ⊢ (𝑣 = 𝐴 → ((𝑣𝑂𝑌) = 𝑍 ↔ (𝐴𝑂𝑌) = 𝑍))
7		suppssov2.o	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑅) → (𝑣𝑂𝑌) = 𝑍)
8	7	ralrimiva 3126	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑣 ∈ 𝑅 (𝑣𝑂𝑌) = 𝑍)
9	8	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥 ∈ 𝐷) → ∀𝑣 ∈ 𝑅 (𝑣𝑂𝑌) = 𝑍)
10		suppssov2.a	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐴 ∈ 𝑅)
11	10	adantlr 715	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥 ∈ 𝐷) → 𝐴 ∈ 𝑅)
12	6, 9, 11	rspcdva 3592	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥 ∈ 𝐷) → (𝐴𝑂𝑌) = 𝑍)
13		oveq2 7397	. . . . . . . . . . 11 ⊢ (𝐵 = 𝑌 → (𝐴𝑂𝐵) = (𝐴𝑂𝑌))
14	13	eqeq1d 2732	. . . . . . . . . 10 ⊢ (𝐵 = 𝑌 → ((𝐴𝑂𝐵) = 𝑍 ↔ (𝐴𝑂𝑌) = 𝑍))
15	12, 14	syl5ibrcom 247	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥 ∈ 𝐷) → (𝐵 = 𝑌 → (𝐴𝑂𝐵) = 𝑍))
16	15	necon3d 2947	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥 ∈ 𝐷) → ((𝐴𝑂𝐵) ≠ 𝑍 → 𝐵 ≠ 𝑌))
17		eldifsni 4756	. . . . . . . 8 ⊢ ((𝐴𝑂𝐵) ∈ (V ∖ {𝑍}) → (𝐴𝑂𝐵) ≠ 𝑍)
18	16, 17	impel 505	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})) → 𝐵 ≠ 𝑌)
19		eldifsn 4752	. . . . . . 7 ⊢ (𝐵 ∈ (V ∖ {𝑌}) ↔ (𝐵 ∈ V ∧ 𝐵 ≠ 𝑌))
20	4, 18, 19	sylanbrc 583	. . . . . 6 ⊢ ((((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})) → 𝐵 ∈ (V ∖ {𝑌}))
21	20	ex 412	. . . . 5 ⊢ (((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥 ∈ 𝐷) → ((𝐴𝑂𝐵) ∈ (V ∖ {𝑍}) → 𝐵 ∈ (V ∖ {𝑌})))
22	21	ss2rabdv 4041	. . . 4 ⊢ ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → {𝑥 ∈ 𝐷 ∣ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})} ⊆ {𝑥 ∈ 𝐷 ∣ 𝐵 ∈ (V ∖ {𝑌})})
23		eqid 2730	. . . . 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 8167	. . . 4 ⊢ ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → ((𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) = {𝑥 ∈ 𝐷 ∣ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})})
27		eqid 2730	. . . . 5 ⊢ (𝑥 ∈ 𝐷 ↦ 𝐵) = (𝑥 ∈ 𝐷 ↦ 𝐵)
28		suppssov2.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑊)
29	28	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → 𝑌 ∈ 𝑊)
30	27, 24, 29	mptsuppdifd 8167	. . . 4 ⊢ ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → ((𝑥 ∈ 𝐷 ↦ 𝐵) supp 𝑌) = {𝑥 ∈ 𝐷 ∣ 𝐵 ∈ (V ∖ {𝑌})})
31	22, 26, 30	3sstr4d 4004	. . 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 480	. . 3 ⊢ ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → ((𝑥 ∈ 𝐷 ↦ 𝐵) supp 𝑌) ⊆ 𝐿)
34	31, 33	sstrd 3959	. 2 ⊢ ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → ((𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) ⊆ 𝐿)
35		mptexg 7197	. . . . . . 7 ⊢ (𝐷 ∈ V → (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) ∈ V)
36		ovex 7422	. . . . . . . . . 10 ⊢ (𝐴𝑂𝐵) ∈ V
37	36	rgenw 3049	. . . . . . . . 9 ⊢ ∀𝑥 ∈ 𝐷 (𝐴𝑂𝐵) ∈ V
38		dmmptg 6217	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐷 (𝐴𝑂𝐵) ∈ V → dom (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) = 𝐷)
39	37, 38	ax-mp 5	. . . . . . . 8 ⊢ dom (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) = 𝐷
40		dmexg 7879	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) ∈ V → dom (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) ∈ V)
41	39, 40	eqeltrrid 2834	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) ∈ V → 𝐷 ∈ V)
42	35, 41	impbii 209	. . . . . 6 ⊢ (𝐷 ∈ V ↔ (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) ∈ V)
43	42	anbi1i 624	. . . . 5 ⊢ ((𝐷 ∈ V ∧ 𝑍 ∈ V) ↔ ((𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) ∈ V ∧ 𝑍 ∈ V))
44		supp0prc 8144	. . . . 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 4365	. . . 4 ⊢ ∅ ⊆ 𝐿
47	45, 46	eqsstrdi 3993	. . 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 1540   ∈ wcel 2109   ≠ wne 2926  ∀wral 3045  {crab 3408  Vcvv 3450   ∖ cdif 3913   ⊆ wss 3916  ∅c0 4298  {csn 4591   ↦ cmpt 5190  dom cdm 5640  (class class class)co 7389   supp csupp 8141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pr 5389  ax-un 7713

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-ov 7392  df-oprab 7393  df-mpo 7394  df-supp 8142

This theorem is referenced by:  psdmplcl  22055