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Theorem suppssov1 8203
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 → ((𝑥𝐷𝐴) supp 𝑌) ⊆ 𝐿)
suppssov1.o ((𝜑𝑣𝑅) → (𝑌𝑂𝑣) = 𝑍)
suppssov1.a ((𝜑𝑥𝐷) → 𝐴𝑉)
suppssov1.b ((𝜑𝑥𝐷) → 𝐵𝑅)
suppssov1.y (𝜑𝑌𝑊)
Assertion
Ref Expression
suppssov1 (𝜑 → ((𝑥𝐷 ↦ (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
StepHypRef Expression
1 suppssov1.a . . . . . . . . . 10 ((𝜑𝑥𝐷) → 𝐴𝑉)
21elexd 3483 . . . . . . . . 9 ((𝜑𝑥𝐷) → 𝐴 ∈ V)
32adantlr 713 . . . . . . . 8 (((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥𝐷) → 𝐴 ∈ V)
43adantr 479 . . . . . . 7 ((((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥𝐷) ∧ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})) → 𝐴 ∈ V)
5 oveq2 7427 . . . . . . . . . . . 12 (𝑣 = 𝐵 → (𝑌𝑂𝑣) = (𝑌𝑂𝐵))
65eqeq1d 2727 . . . . . . . . . . 11 (𝑣 = 𝐵 → ((𝑌𝑂𝑣) = 𝑍 ↔ (𝑌𝑂𝐵) = 𝑍))
7 suppssov1.o . . . . . . . . . . . . 13 ((𝜑𝑣𝑅) → (𝑌𝑂𝑣) = 𝑍)
87ralrimiva 3135 . . . . . . . . . . . 12 (𝜑 → ∀𝑣𝑅 (𝑌𝑂𝑣) = 𝑍)
98ad2antrr 724 . . . . . . . . . . 11 (((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥𝐷) → ∀𝑣𝑅 (𝑌𝑂𝑣) = 𝑍)
10 suppssov1.b . . . . . . . . . . . 12 ((𝜑𝑥𝐷) → 𝐵𝑅)
1110adantlr 713 . . . . . . . . . . 11 (((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥𝐷) → 𝐵𝑅)
126, 9, 11rspcdva 3607 . . . . . . . . . 10 (((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥𝐷) → (𝑌𝑂𝐵) = 𝑍)
13 oveq1 7426 . . . . . . . . . . 11 (𝐴 = 𝑌 → (𝐴𝑂𝐵) = (𝑌𝑂𝐵))
1413eqeq1d 2727 . . . . . . . . . 10 (𝐴 = 𝑌 → ((𝐴𝑂𝐵) = 𝑍 ↔ (𝑌𝑂𝐵) = 𝑍))
1512, 14syl5ibrcom 246 . . . . . . . . 9 (((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥𝐷) → (𝐴 = 𝑌 → (𝐴𝑂𝐵) = 𝑍))
1615necon3d 2950 . . . . . . . 8 (((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥𝐷) → ((𝐴𝑂𝐵) ≠ 𝑍𝐴𝑌))
17 eldifsni 4795 . . . . . . . 8 ((𝐴𝑂𝐵) ∈ (V ∖ {𝑍}) → (𝐴𝑂𝐵) ≠ 𝑍)
1816, 17impel 504 . . . . . . 7 ((((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥𝐷) ∧ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})) → 𝐴𝑌)
19 eldifsn 4792 . . . . . . 7 (𝐴 ∈ (V ∖ {𝑌}) ↔ (𝐴 ∈ V ∧ 𝐴𝑌))
204, 18, 19sylanbrc 581 . . . . . 6 ((((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥𝐷) ∧ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})) → 𝐴 ∈ (V ∖ {𝑌}))
2120ex 411 . . . . 5 (((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) ∧ 𝑥𝐷) → ((𝐴𝑂𝐵) ∈ (V ∖ {𝑍}) → 𝐴 ∈ (V ∖ {𝑌})))
2221ss2rabdv 4069 . . . 4 ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → {𝑥𝐷 ∣ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})} ⊆ {𝑥𝐷𝐴 ∈ (V ∖ {𝑌})})
23 eqid 2725 . . . . 5 (𝑥𝐷 ↦ (𝐴𝑂𝐵)) = (𝑥𝐷 ↦ (𝐴𝑂𝐵))
24 simprl 769 . . . . 5 ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → 𝐷 ∈ V)
25 simprr 771 . . . . 5 ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → 𝑍 ∈ V)
2623, 24, 25mptsuppdifd 8191 . . . 4 ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → ((𝑥𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) = {𝑥𝐷 ∣ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})})
27 eqid 2725 . . . . 5 (𝑥𝐷𝐴) = (𝑥𝐷𝐴)
28 suppssov1.y . . . . . 6 (𝜑𝑌𝑊)
2928adantr 479 . . . . 5 ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → 𝑌𝑊)
3027, 24, 29mptsuppdifd 8191 . . . 4 ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → ((𝑥𝐷𝐴) supp 𝑌) = {𝑥𝐷𝐴 ∈ (V ∖ {𝑌})})
3122, 26, 303sstr4d 4024 . . 3 ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → ((𝑥𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) ⊆ ((𝑥𝐷𝐴) supp 𝑌))
32 suppssov1.s . . . 4 (𝜑 → ((𝑥𝐷𝐴) supp 𝑌) ⊆ 𝐿)
3332adantr 479 . . 3 ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → ((𝑥𝐷𝐴) supp 𝑌) ⊆ 𝐿)
3431, 33sstrd 3987 . 2 ((𝜑 ∧ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → ((𝑥𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) ⊆ 𝐿)
35 mptexg 7233 . . . . . . 7 (𝐷 ∈ V → (𝑥𝐷 ↦ (𝐴𝑂𝐵)) ∈ V)
36 ovex 7452 . . . . . . . . . 10 (𝐴𝑂𝐵) ∈ V
3736rgenw 3054 . . . . . . . . 9 𝑥𝐷 (𝐴𝑂𝐵) ∈ V
38 dmmptg 6248 . . . . . . . . 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 2830 . . . . . . 7 ((𝑥𝐷 ↦ (𝐴𝑂𝐵)) ∈ V → 𝐷 ∈ V)
4235, 41impbii 208 . . . . . 6 (𝐷 ∈ V ↔ (𝑥𝐷 ↦ (𝐴𝑂𝐵)) ∈ V)
4342anbi1i 622 . . . . 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 329 . . . 4 (¬ (𝐷 ∈ V ∧ 𝑍 ∈ V) → ((𝑥𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) = ∅)
46 0ss 4398 . . . 4 ∅ ⊆ 𝐿
4745, 46eqsstrdi 4031 . . 3 (¬ (𝐷 ∈ V ∧ 𝑍 ∈ V) → ((𝑥𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) ⊆ 𝐿)
4847adantl 480 . 2 ((𝜑 ∧ ¬ (𝐷 ∈ V ∧ 𝑍 ∈ V)) → ((𝑥𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) ⊆ 𝐿)
4934, 48pm2.61dan 811 1 (𝜑 → ((𝑥𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) ⊆ 𝐿)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394   = wceq 1533  wcel 2098  wne 2929  wral 3050  {crab 3418  Vcvv 3461  cdif 3941  wss 3944  c0 4322  {csn 4630  cmpt 5232  dom cdm 5678  (class class class)co 7419   supp csupp 8165
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  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 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-supp 8166
This theorem is referenced by:  suppssof1  8205  fsuppssov1  9409  evlslem6  22049  plypf1  26191  mhphf  41965
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