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Theorem suppssof1 5670
Description: Formula building theorem for support restrictions: vector operation with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.)
Hypotheses
Ref Expression
suppssof1.s (φ → (A “ (V ∖ {𝑌})) ⊆ 𝐿)
suppssof1.o ((φ v 𝑅) → (𝑌𝑂v) = 𝑍)
suppssof1.a (φA:𝐷𝑉)
suppssof1.b (φB:𝐷𝑅)
suppssof1.d (φ𝐷 𝑊)
Assertion
Ref Expression
suppssof1 (φ → ((A𝑓 𝑂B) “ (V ∖ {𝑍})) ⊆ 𝐿)
Distinct variable groups:   φ,v   v,B   v,𝑂   v,𝑅   v,𝑌   v,𝑍
Allowed substitution hints:   A(v)   𝐷(v)   𝐿(v)   𝑉(v)   𝑊(v)

Proof of Theorem suppssof1
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 suppssof1.a . . . . . 6 (φA:𝐷𝑉)
2 ffn 4989 . . . . . 6 (A:𝐷𝑉A Fn 𝐷)
31, 2syl 14 . . . . 5 (φA Fn 𝐷)
4 suppssof1.b . . . . . 6 (φB:𝐷𝑅)
5 ffn 4989 . . . . . 6 (B:𝐷𝑅B Fn 𝐷)
64, 5syl 14 . . . . 5 (φB Fn 𝐷)
7 suppssof1.d . . . . 5 (φ𝐷 𝑊)
8 inidm 3140 . . . . 5 (𝐷𝐷) = 𝐷
9 eqidd 2038 . . . . 5 ((φ x 𝐷) → (Ax) = (Ax))
10 eqidd 2038 . . . . 5 ((φ x 𝐷) → (Bx) = (Bx))
113, 6, 7, 7, 8, 9, 10offval 5661 . . . 4 (φ → (A𝑓 𝑂B) = (x 𝐷 ↦ ((Ax)𝑂(Bx))))
1211cnveqd 4454 . . 3 (φ(A𝑓 𝑂B) = (x 𝐷 ↦ ((Ax)𝑂(Bx))))
1312imaeq1d 4610 . 2 (φ → ((A𝑓 𝑂B) “ (V ∖ {𝑍})) = ((x 𝐷 ↦ ((Ax)𝑂(Bx))) “ (V ∖ {𝑍})))
141feqmptd 5169 . . . . . 6 (φA = (x 𝐷 ↦ (Ax)))
1514cnveqd 4454 . . . . 5 (φA = (x 𝐷 ↦ (Ax)))
1615imaeq1d 4610 . . . 4 (φ → (A “ (V ∖ {𝑌})) = ((x 𝐷 ↦ (Ax)) “ (V ∖ {𝑌})))
17 suppssof1.s . . . 4 (φ → (A “ (V ∖ {𝑌})) ⊆ 𝐿)
1816, 17eqsstr3d 2974 . . 3 (φ → ((x 𝐷 ↦ (Ax)) “ (V ∖ {𝑌})) ⊆ 𝐿)
19 suppssof1.o . . 3 ((φ v 𝑅) → (𝑌𝑂v) = 𝑍)
20 funfvex 5135 . . . . 5 ((Fun A x dom A) → (Ax) V)
2120funfni 4942 . . . 4 ((A Fn 𝐷 x 𝐷) → (Ax) V)
223, 21sylan 267 . . 3 ((φ x 𝐷) → (Ax) V)
234ffvelrnda 5245 . . 3 ((φ x 𝐷) → (Bx) 𝑅)
2418, 19, 22, 23suppssov1 5651 . 2 (φ → ((x 𝐷 ↦ ((Ax)𝑂(Bx))) “ (V ∖ {𝑍})) ⊆ 𝐿)
2513, 24eqsstrd 2973 1 (φ → ((A𝑓 𝑂B) “ (V ∖ {𝑍})) ⊆ 𝐿)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  Vcvv 2551  cdif 2908  wss 2911  {csn 3367  cmpt 3809  ccnv 4287  cima 4291   Fn wfn 4840  wf 4841  cfv 4845  (class class class)co 5455  𝑓 cof 5652
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-setind 4220
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-of 5654
This theorem is referenced by: (None)
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