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Theorem suppssov2 8197
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 3490 . . . . . . . . 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 7421 . . . . . . . . . . . 12 (𝑣 = 𝐴 → (𝑣𝑂𝑌) = (𝐴𝑂𝑌))
65eqeq1d 2729 . . . . . . . . . . 11 (𝑣 = 𝐴 → ((𝑣𝑂𝑌) = 𝑍 ↔ (𝐴𝑂𝑌) = 𝑍))
7 suppssov2.o . . . . . . . . . . . . 13 ((𝜑𝑣𝑅) → (𝑣𝑂𝑌) = 𝑍)
87ralrimiva 3141 . . . . . . . . . . . 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 3608 . . . . . . . . . 10 (((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥𝐷) → (𝐴𝑂𝑌) = 𝑍)
13 oveq2 7422 . . . . . . . . . . 11 (𝐵 = 𝑌 → (𝐴𝑂𝐵) = (𝐴𝑂𝑌))
1413eqeq1d 2729 . . . . . . . . . 10 (𝐵 = 𝑌 → ((𝐴𝑂𝐵) = 𝑍 ↔ (𝐴𝑂𝑌) = 𝑍))
1512, 14syl5ibrcom 246 . . . . . . . . 9 (((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥𝐷) → (𝐵 = 𝑌 → (𝐴𝑂𝐵) = 𝑍))
1615necon3d 2956 . . . . . . . 8 (((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥𝐷) → ((𝐴𝑂𝐵) ≠ 𝑍𝐵𝑌))
17 eldifsni 4789 . . . . . . . 8 ((𝐴𝑂𝐵) ∈ (V ∖ {𝑍}) → (𝐴𝑂𝐵) ≠ 𝑍)
1816, 17impel 505 . . . . . . 7 ((((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥𝐷) ∧ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})) → 𝐵𝑌)
19 eldifsn 4786 . . . . . . 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 4069 . . . 4 ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → {𝑥𝐷 ∣ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})} ⊆ {𝑥𝐷𝐵 ∈ (V ∖ {𝑌})})
23 eqid 2727 . . . . 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 8184 . . . 4 ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → ((𝑥𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) = {𝑥𝐷 ∣ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})})
27 eqid 2727 . . . . 5 (𝑥𝐷𝐵) = (𝑥𝐷𝐵)
28 suppssov2.y . . . . . 6 (𝜑𝑌𝑊)
2928adantr 480 . . . . 5 ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → 𝑌𝑊)
3027, 24, 29mptsuppdifd 8184 . . . 4 ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → ((𝑥𝐷𝐵) supp 𝑌) = {𝑥𝐷𝐵 ∈ (V ∖ {𝑌})})
3122, 26, 303sstr4d 4025 . . 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 3988 . 2 ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → ((𝑥𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) ⊆ 𝐿)
35 mptexg 7227 . . . . . . 7 (𝐷 ∈ V → (𝑥𝐷 ↦ (𝐴𝑂𝐵)) ∈ V)
36 ovex 7447 . . . . . . . . . 10 (𝐴𝑂𝐵) ∈ V
3736rgenw 3060 . . . . . . . . 9 𝑥𝐷 (𝐴𝑂𝐵) ∈ V
38 dmmptg 6240 . . . . . . . . 9 (∀𝑥𝐷 (𝐴𝑂𝐵) ∈ V → dom (𝑥𝐷 ↦ (𝐴𝑂𝐵)) = 𝐷)
3937, 38ax-mp 5 . . . . . . . 8 dom (𝑥𝐷 ↦ (𝐴𝑂𝐵)) = 𝐷
40 dmexg 7903 . . . . . . . 8 ((𝑥𝐷 ↦ (𝐴𝑂𝐵)) ∈ V → dom (𝑥𝐷 ↦ (𝐴𝑂𝐵)) ∈ V)
4139, 40eqeltrrid 2833 . . . . . . 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 8162 . . . . 5 (¬ ((𝑥𝐷 ↦ (𝐴𝑂𝐵)) ∈ V ∧ 𝑍 ∈ V) → ((𝑥𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) = ∅)
4543, 44sylnbi 330 . . . 4 (¬ (𝐷 ∈ V ∧ 𝑍 ∈ V) → ((𝑥𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) = ∅)
46 0ss 4392 . . . 4 ∅ ⊆ 𝐿
4745, 46eqsstrdi 4032 . . 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 2935  wral 3056  {crab 3427  Vcvv 3469  cdif 3941  wss 3944  c0 4318  {csn 4624  cmpt 5225  dom cdm 5672  (class class class)co 7414   supp csupp 8159
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 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734
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 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-supp 8160
This theorem is referenced by:  psdmplcl  22073
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