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Theorem suppssov2 8203
Description: Formula building theorem for support restrictions: operator with right annihilator. (Contributed by SN, 11-Apr-2025.)
Hypotheses
Ref Expression
suppssov2.s (𝜑 → ((𝑥𝐷𝐵) supp 𝑌) ⊆ 𝐿)
suppssov2.o ((𝜑𝑣𝑅) → (𝑣𝑂𝑌) = 𝑍)
suppssov2.a ((𝜑𝑥𝐷) → 𝐴𝑅)
suppssov2.b ((𝜑𝑥𝐷) → 𝐵𝑉)
suppssov2.y (𝜑𝑌𝑊)
Assertion
Ref Expression
suppssov2 (𝜑 → ((𝑥𝐷 ↦ (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
StepHypRef Expression
1 suppssov2.b . . . . . . . . . 10 ((𝜑𝑥𝐷) → 𝐵𝑉)
21elexd 3492 . . . . . . . . 9 ((𝜑𝑥𝐷) → 𝐵 ∈ V)
32adantlr 714 . . . . . . . 8 (((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥𝐷) → 𝐵 ∈ V)
43adantr 480 . . . . . . 7 ((((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥𝐷) ∧ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})) → 𝐵 ∈ V)
5 oveq1 7427 . . . . . . . . . . . 12 (𝑣 = 𝐴 → (𝑣𝑂𝑌) = (𝐴𝑂𝑌))
65eqeq1d 2730 . . . . . . . . . . 11 (𝑣 = 𝐴 → ((𝑣𝑂𝑌) = 𝑍 ↔ (𝐴𝑂𝑌) = 𝑍))
7 suppssov2.o . . . . . . . . . . . . 13 ((𝜑𝑣𝑅) → (𝑣𝑂𝑌) = 𝑍)
87ralrimiva 3143 . . . . . . . . . . . 12 (𝜑 → ∀𝑣𝑅 (𝑣𝑂𝑌) = 𝑍)
98ad2antrr 725 . . . . . . . . . . 11 (((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥𝐷) → ∀𝑣𝑅 (𝑣𝑂𝑌) = 𝑍)
10 suppssov2.a . . . . . . . . . . . 12 ((𝜑𝑥𝐷) → 𝐴𝑅)
1110adantlr 714 . . . . . . . . . . 11 (((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥𝐷) → 𝐴𝑅)
126, 9, 11rspcdva 3610 . . . . . . . . . 10 (((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥𝐷) → (𝐴𝑂𝑌) = 𝑍)
13 oveq2 7428 . . . . . . . . . . 11 (𝐵 = 𝑌 → (𝐴𝑂𝐵) = (𝐴𝑂𝑌))
1413eqeq1d 2730 . . . . . . . . . 10 (𝐵 = 𝑌 → ((𝐴𝑂𝐵) = 𝑍 ↔ (𝐴𝑂𝑌) = 𝑍))
1512, 14syl5ibrcom 246 . . . . . . . . 9 (((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥𝐷) → (𝐵 = 𝑌 → (𝐴𝑂𝐵) = 𝑍))
1615necon3d 2958 . . . . . . . 8 (((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥𝐷) → ((𝐴𝑂𝐵) ≠ 𝑍𝐵𝑌))
17 eldifsni 4794 . . . . . . . 8 ((𝐴𝑂𝐵) ∈ (V ∖ {𝑍}) → (𝐴𝑂𝐵) ≠ 𝑍)
1816, 17impel 505 . . . . . . 7 ((((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥𝐷) ∧ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})) → 𝐵𝑌)
19 eldifsn 4791 . . . . . . 7 (𝐵 ∈ (V ∖ {𝑌}) ↔ (𝐵 ∈ V ∧ 𝐵𝑌))
204, 18, 19sylanbrc 582 . . . . . 6 ((((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥𝐷) ∧ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})) → 𝐵 ∈ (V ∖ {𝑌}))
2120ex 412 . . . . 5 (((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥𝐷) → ((𝐴𝑂𝐵) ∈ (V ∖ {𝑍}) → 𝐵 ∈ (V ∖ {𝑌})))
2221ss2rabdv 4071 . . . 4 ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → {𝑥𝐷 ∣ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})} ⊆ {𝑥𝐷𝐵 ∈ (V ∖ {𝑌})})
23 eqid 2728 . . . . 5 (𝑥𝐷 ↦ (𝐴𝑂𝐵)) = (𝑥𝐷 ↦ (𝐴𝑂𝐵))
24 simprl 770 . . . . 5 ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → 𝐷 ∈ V)
25 simprr 772 . . . . 5 ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → 𝑍 ∈ V)
2623, 24, 25mptsuppdifd 8190 . . . 4 ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → ((𝑥𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) = {𝑥𝐷 ∣ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})})
27 eqid 2728 . . . . 5 (𝑥𝐷𝐵) = (𝑥𝐷𝐵)
28 suppssov2.y . . . . . 6 (𝜑𝑌𝑊)
2928adantr 480 . . . . 5 ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → 𝑌𝑊)
3027, 24, 29mptsuppdifd 8190 . . . 4 ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → ((𝑥𝐷𝐵) supp 𝑌) = {𝑥𝐷𝐵 ∈ (V ∖ {𝑌})})
3122, 26, 303sstr4d 4027 . . 3 ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → ((𝑥𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) ⊆ ((𝑥𝐷𝐵) supp 𝑌))
32 suppssov2.s . . . 4 (𝜑 → ((𝑥𝐷𝐵) supp 𝑌) ⊆ 𝐿)
3332adantr 480 . . 3 ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → ((𝑥𝐷𝐵) supp 𝑌) ⊆ 𝐿)
3431, 33sstrd 3990 . 2 ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → ((𝑥𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) ⊆ 𝐿)
35 mptexg 7233 . . . . . . 7 (𝐷 ∈ V → (𝑥𝐷 ↦ (𝐴𝑂𝐵)) ∈ V)
36 ovex 7453 . . . . . . . . . 10 (𝐴𝑂𝐵) ∈ V
3736rgenw 3062 . . . . . . . . 9 𝑥𝐷 (𝐴𝑂𝐵) ∈ V
38 dmmptg 6246 . . . . . . . . 9 (∀𝑥𝐷 (𝐴𝑂𝐵) ∈ V → dom (𝑥𝐷 ↦ (𝐴𝑂𝐵)) = 𝐷)
3937, 38ax-mp 5 . . . . . . . 8 dom (𝑥𝐷 ↦ (𝐴𝑂𝐵)) = 𝐷
40 dmexg 7909 . . . . . . . 8 ((𝑥𝐷 ↦ (𝐴𝑂𝐵)) ∈ V → dom (𝑥𝐷 ↦ (𝐴𝑂𝐵)) ∈ V)
4139, 40eqeltrrid 2834 . . . . . . 7 ((𝑥𝐷 ↦ (𝐴𝑂𝐵)) ∈ V → 𝐷 ∈ V)
4235, 41impbii 208 . . . . . 6 (𝐷 ∈ V ↔ (𝑥𝐷 ↦ (𝐴𝑂𝐵)) ∈ V)
4342anbi1i 623 . . . . 5 ((𝐷 ∈ V ∧ 𝑍 ∈ V) ↔ ((𝑥𝐷 ↦ (𝐴𝑂𝐵)) ∈ V ∧ 𝑍 ∈ V))
44 supp0prc 8168 . . . . 5 (¬ ((𝑥𝐷 ↦ (𝐴𝑂𝐵)) ∈ V ∧ 𝑍 ∈ V) → ((𝑥𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) = ∅)
4543, 44sylnbi 330 . . . 4 (¬ (𝐷 ∈ V ∧ 𝑍 ∈ V) → ((𝑥𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) = ∅)
46 0ss 4397 . . . 4 ∅ ⊆ 𝐿
4745, 46eqsstrdi 4034 . . 3 (¬ (𝐷 ∈ V ∧ 𝑍 ∈ V) → ((𝑥𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) ⊆ 𝐿)
4847adantl 481 . 2 ((𝜑 ∧ ¬ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → ((𝑥𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) ⊆ 𝐿)
4934, 48pm2.61dan 812 1 (𝜑 → ((𝑥𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) ⊆ 𝐿)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1534  wcel 2099  wne 2937  wral 3058  {crab 3429  Vcvv 3471  cdif 3944  wss 3947  c0 4323  {csn 4629  cmpt 5231  dom cdm 5678  (class class class)co 7420   supp csupp 8165
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-supp 8166
This theorem is referenced by:  psdmplcl  22085
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