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Theorem suppssof1 5756
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 ((𝜑𝑣𝑅) → (𝑌𝑂𝑣) = 𝑍)
suppssof1.a (𝜑𝐴:𝐷𝑉)
suppssof1.b (𝜑𝐵:𝐷𝑅)
suppssof1.d (𝜑𝐷𝑊)
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 5074 . . . . . 6 (𝐴:𝐷𝑉𝐴 Fn 𝐷)
31, 2syl 14 . . . . 5 (𝜑𝐴 Fn 𝐷)
4 suppssof1.b . . . . . 6 (𝜑𝐵:𝐷𝑅)
5 ffn 5074 . . . . . 6 (𝐵:𝐷𝑅𝐵 Fn 𝐷)
64, 5syl 14 . . . . 5 (𝜑𝐵 Fn 𝐷)
7 suppssof1.d . . . . 5 (𝜑𝐷𝑊)
8 inidm 3174 . . . . 5 (𝐷𝐷) = 𝐷
9 eqidd 2057 . . . . 5 ((𝜑𝑥𝐷) → (𝐴𝑥) = (𝐴𝑥))
10 eqidd 2057 . . . . 5 ((𝜑𝑥𝐷) → (𝐵𝑥) = (𝐵𝑥))
113, 6, 7, 7, 8, 9, 10offval 5747 . . . 4 (𝜑 → (𝐴𝑓 𝑂𝐵) = (𝑥𝐷 ↦ ((𝐴𝑥)𝑂(𝐵𝑥))))
1211cnveqd 4539 . . 3 (𝜑(𝐴𝑓 𝑂𝐵) = (𝑥𝐷 ↦ ((𝐴𝑥)𝑂(𝐵𝑥))))
1312imaeq1d 4695 . 2 (𝜑 → ((𝐴𝑓 𝑂𝐵) “ (V ∖ {𝑍})) = ((𝑥𝐷 ↦ ((𝐴𝑥)𝑂(𝐵𝑥))) “ (V ∖ {𝑍})))
141feqmptd 5254 . . . . . 6 (𝜑𝐴 = (𝑥𝐷 ↦ (𝐴𝑥)))
1514cnveqd 4539 . . . . 5 (𝜑𝐴 = (𝑥𝐷 ↦ (𝐴𝑥)))
1615imaeq1d 4695 . . . 4 (𝜑 → (𝐴 “ (V ∖ {𝑌})) = ((𝑥𝐷 ↦ (𝐴𝑥)) “ (V ∖ {𝑌})))
17 suppssof1.s . . . 4 (𝜑 → (𝐴 “ (V ∖ {𝑌})) ⊆ 𝐿)
1816, 17eqsstr3d 3008 . . 3 (𝜑 → ((𝑥𝐷 ↦ (𝐴𝑥)) “ (V ∖ {𝑌})) ⊆ 𝐿)
19 suppssof1.o . . 3 ((𝜑𝑣𝑅) → (𝑌𝑂𝑣) = 𝑍)
20 funfvex 5220 . . . . 5 ((Fun 𝐴𝑥 ∈ dom 𝐴) → (𝐴𝑥) ∈ V)
2120funfni 5027 . . . 4 ((𝐴 Fn 𝐷𝑥𝐷) → (𝐴𝑥) ∈ V)
223, 21sylan 271 . . 3 ((𝜑𝑥𝐷) → (𝐴𝑥) ∈ V)
234ffvelrnda 5330 . . 3 ((𝜑𝑥𝐷) → (𝐵𝑥) ∈ 𝑅)
2418, 19, 22, 23suppssov1 5737 . 2 (𝜑 → ((𝑥𝐷 ↦ ((𝐴𝑥)𝑂(𝐵𝑥))) “ (V ∖ {𝑍})) ⊆ 𝐿)
2513, 24eqsstrd 3007 1 (𝜑 → ((𝐴𝑓 𝑂𝐵) “ (V ∖ {𝑍})) ⊆ 𝐿)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wcel 1409  Vcvv 2574  cdif 2942  wss 2945  {csn 3403  cmpt 3846  ccnv 4372  cima 4376   Fn wfn 4925  wf 4926  cfv 4930  (class class class)co 5540  𝑓 cof 5738
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-setind 4290
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-of 5740
This theorem is referenced by: (None)
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