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Theorem suppssof1 5910
 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 5195 . . . . . 6 (𝐴:𝐷𝑉𝐴 Fn 𝐷)
31, 2syl 14 . . . . 5 (𝜑𝐴 Fn 𝐷)
4 suppssof1.b . . . . . 6 (𝜑𝐵:𝐷𝑅)
5 ffn 5195 . . . . . 6 (𝐵:𝐷𝑅𝐵 Fn 𝐷)
64, 5syl 14 . . . . 5 (𝜑𝐵 Fn 𝐷)
7 suppssof1.d . . . . 5 (𝜑𝐷𝑊)
8 inidm 3224 . . . . 5 (𝐷𝐷) = 𝐷
9 eqidd 2096 . . . . 5 ((𝜑𝑥𝐷) → (𝐴𝑥) = (𝐴𝑥))
10 eqidd 2096 . . . . 5 ((𝜑𝑥𝐷) → (𝐵𝑥) = (𝐵𝑥))
113, 6, 7, 7, 8, 9, 10offval 5901 . . . 4 (𝜑 → (𝐴𝑓 𝑂𝐵) = (𝑥𝐷 ↦ ((𝐴𝑥)𝑂(𝐵𝑥))))
1211cnveqd 4643 . . 3 (𝜑(𝐴𝑓 𝑂𝐵) = (𝑥𝐷 ↦ ((𝐴𝑥)𝑂(𝐵𝑥))))
1312imaeq1d 4806 . 2 (𝜑 → ((𝐴𝑓 𝑂𝐵) “ (V ∖ {𝑍})) = ((𝑥𝐷 ↦ ((𝐴𝑥)𝑂(𝐵𝑥))) “ (V ∖ {𝑍})))
141feqmptd 5392 . . . . . 6 (𝜑𝐴 = (𝑥𝐷 ↦ (𝐴𝑥)))
1514cnveqd 4643 . . . . 5 (𝜑𝐴 = (𝑥𝐷 ↦ (𝐴𝑥)))
1615imaeq1d 4806 . . . 4 (𝜑 → (𝐴 “ (V ∖ {𝑌})) = ((𝑥𝐷 ↦ (𝐴𝑥)) “ (V ∖ {𝑌})))
17 suppssof1.s . . . 4 (𝜑 → (𝐴 “ (V ∖ {𝑌})) ⊆ 𝐿)
1816, 17eqsstr3d 3076 . . 3 (𝜑 → ((𝑥𝐷 ↦ (𝐴𝑥)) “ (V ∖ {𝑌})) ⊆ 𝐿)
19 suppssof1.o . . 3 ((𝜑𝑣𝑅) → (𝑌𝑂𝑣) = 𝑍)
20 funfvex 5357 . . . . 5 ((Fun 𝐴𝑥 ∈ dom 𝐴) → (𝐴𝑥) ∈ V)
2120funfni 5148 . . . 4 ((𝐴 Fn 𝐷𝑥𝐷) → (𝐴𝑥) ∈ V)
223, 21sylan 278 . . 3 ((𝜑𝑥𝐷) → (𝐴𝑥) ∈ V)
234ffvelrnda 5473 . . 3 ((𝜑𝑥𝐷) → (𝐵𝑥) ∈ 𝑅)
2418, 19, 22, 23suppssov1 5891 . 2 (𝜑 → ((𝑥𝐷 ↦ ((𝐴𝑥)𝑂(𝐵𝑥))) “ (V ∖ {𝑍})) ⊆ 𝐿)
2513, 24eqsstrd 3075 1 (𝜑 → ((𝐴𝑓 𝑂𝐵) “ (V ∖ {𝑍})) ⊆ 𝐿)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1296   ∈ wcel 1445  Vcvv 2633   ∖ cdif 3010   ⊆ wss 3013  {csn 3466   ↦ cmpt 3921  ◡ccnv 4466   “ cima 4470   Fn wfn 5044  ⟶wf 5045  ‘cfv 5049  (class class class)co 5690   ∘𝑓 cof 5892 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-setind 4381 This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-of 5894 This theorem is referenced by: (None)
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