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Theorem suppssof1 5647
 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 4968 . . . . . 6 (A:𝐷𝑉A Fn 𝐷)
31, 2syl 14 . . . . 5 (φA Fn 𝐷)
4 suppssof1.b . . . . . 6 (φB:𝐷𝑅)
5 ffn 4968 . . . . . 6 (B:𝐷𝑅B Fn 𝐷)
64, 5syl 14 . . . . 5 (φB Fn 𝐷)
7 suppssof1.d . . . . 5 (φ𝐷 𝑊)
8 inidm 3119 . . . . 5 (𝐷𝐷) = 𝐷
9 eqidd 2019 . . . . 5 ((φ x 𝐷) → (Ax) = (Ax))
10 eqidd 2019 . . . . 5 ((φ x 𝐷) → (Bx) = (Bx))
113, 6, 7, 7, 8, 9, 10offval 5638 . . . 4 (φ → (A𝑓 𝑂B) = (x 𝐷 ↦ ((Ax)𝑂(Bx))))
1211cnveqd 4434 . . 3 (φ(A𝑓 𝑂B) = (x 𝐷 ↦ ((Ax)𝑂(Bx))))
1312imaeq1d 4590 . 2 (φ → ((A𝑓 𝑂B) “ (V ∖ {𝑍})) = ((x 𝐷 ↦ ((Ax)𝑂(Bx))) “ (V ∖ {𝑍})))
141feqmptd 5147 . . . . . 6 (φA = (x 𝐷 ↦ (Ax)))
1514cnveqd 4434 . . . . 5 (φA = (x 𝐷 ↦ (Ax)))
1615imaeq1d 4590 . . . 4 (φ → (A “ (V ∖ {𝑌})) = ((x 𝐷 ↦ (Ax)) “ (V ∖ {𝑌})))
17 suppssof1.s . . . 4 (φ → (A “ (V ∖ {𝑌})) ⊆ 𝐿)
1816, 17eqsstr3d 2953 . . 3 (φ → ((x 𝐷 ↦ (Ax)) “ (V ∖ {𝑌})) ⊆ 𝐿)
19 suppssof1.o . . 3 ((φ v 𝑅) → (𝑌𝑂v) = 𝑍)
20 funfvex 5113 . . . . 5 ((Fun A x dom A) → (Ax) V)
2120funfni 4921 . . . 4 ((A Fn 𝐷 x 𝐷) → (Ax) V)
223, 21sylan 267 . . 3 ((φ x 𝐷) → (Ax) V)
234ffvelrnda 5223 . . 3 ((φ x 𝐷) → (Bx) 𝑅)
2418, 19, 22, 23suppssov1 5628 . 2 (φ → ((x 𝐷 ↦ ((Ax)𝑂(Bx))) “ (V ∖ {𝑍})) ⊆ 𝐿)
2513, 24eqsstrd 2952 1 (φ → ((A𝑓 𝑂B) “ (V ∖ {𝑍})) ⊆ 𝐿)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1226   ∈ wcel 1370  Vcvv 2531   ∖ cdif 2887   ⊆ wss 2890  {csn 3346   ↦ cmpt 3788  ◡ccnv 4267   “ cima 4271   Fn wfn 4820  ⟶wf 4821  ‘cfv 4825  (class class class)co 5432   ∘𝑓 cof 5629 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 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-coll 3842  ax-sep 3845  ax-pow 3897  ax-pr 3914  ax-setind 4200 This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-fal 1232  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ne 2184  df-ral 2285  df-rex 2286  df-reu 2287  df-rab 2289  df-v 2533  df-sbc 2738  df-csb 2826  df-dif 2893  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-iun 3629  df-br 3735  df-opab 3789  df-mpt 3790  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-iota 4790  df-fun 4827  df-fn 4828  df-f 4829  df-f1 4830  df-fo 4831  df-f1o 4832  df-fv 4833  df-ov 5435  df-oprab 5436  df-mpt2 5437  df-of 5631 This theorem is referenced by: (None)
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