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Theorem suppssof1 6062
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 5332 . . . . . 6 (𝐴:𝐷𝑉𝐴 Fn 𝐷)
31, 2syl 14 . . . . 5 (𝜑𝐴 Fn 𝐷)
4 suppssof1.b . . . . . 6 (𝜑𝐵:𝐷𝑅)
5 ffn 5332 . . . . . 6 (𝐵:𝐷𝑅𝐵 Fn 𝐷)
64, 5syl 14 . . . . 5 (𝜑𝐵 Fn 𝐷)
7 suppssof1.d . . . . 5 (𝜑𝐷𝑊)
8 inidm 3327 . . . . 5 (𝐷𝐷) = 𝐷
9 eqidd 2165 . . . . 5 ((𝜑𝑥𝐷) → (𝐴𝑥) = (𝐴𝑥))
10 eqidd 2165 . . . . 5 ((𝜑𝑥𝐷) → (𝐵𝑥) = (𝐵𝑥))
113, 6, 7, 7, 8, 9, 10offval 6052 . . . 4 (𝜑 → (𝐴𝑓 𝑂𝐵) = (𝑥𝐷 ↦ ((𝐴𝑥)𝑂(𝐵𝑥))))
1211cnveqd 4775 . . 3 (𝜑(𝐴𝑓 𝑂𝐵) = (𝑥𝐷 ↦ ((𝐴𝑥)𝑂(𝐵𝑥))))
1312imaeq1d 4940 . 2 (𝜑 → ((𝐴𝑓 𝑂𝐵) “ (V ∖ {𝑍})) = ((𝑥𝐷 ↦ ((𝐴𝑥)𝑂(𝐵𝑥))) “ (V ∖ {𝑍})))
141feqmptd 5534 . . . . . 6 (𝜑𝐴 = (𝑥𝐷 ↦ (𝐴𝑥)))
1514cnveqd 4775 . . . . 5 (𝜑𝐴 = (𝑥𝐷 ↦ (𝐴𝑥)))
1615imaeq1d 4940 . . . 4 (𝜑 → (𝐴 “ (V ∖ {𝑌})) = ((𝑥𝐷 ↦ (𝐴𝑥)) “ (V ∖ {𝑌})))
17 suppssof1.s . . . 4 (𝜑 → (𝐴 “ (V ∖ {𝑌})) ⊆ 𝐿)
1816, 17eqsstrrd 3175 . . 3 (𝜑 → ((𝑥𝐷 ↦ (𝐴𝑥)) “ (V ∖ {𝑌})) ⊆ 𝐿)
19 suppssof1.o . . 3 ((𝜑𝑣𝑅) → (𝑌𝑂𝑣) = 𝑍)
20 funfvex 5498 . . . . 5 ((Fun 𝐴𝑥 ∈ dom 𝐴) → (𝐴𝑥) ∈ V)
2120funfni 5283 . . . 4 ((𝐴 Fn 𝐷𝑥𝐷) → (𝐴𝑥) ∈ V)
223, 21sylan 281 . . 3 ((𝜑𝑥𝐷) → (𝐴𝑥) ∈ V)
234ffvelrnda 5615 . . 3 ((𝜑𝑥𝐷) → (𝐵𝑥) ∈ 𝑅)
2418, 19, 22, 23suppssov1 6042 . 2 (𝜑 → ((𝑥𝐷 ↦ ((𝐴𝑥)𝑂(𝐵𝑥))) “ (V ∖ {𝑍})) ⊆ 𝐿)
2513, 24eqsstrd 3174 1 (𝜑 → ((𝐴𝑓 𝑂𝐵) “ (V ∖ {𝑍})) ⊆ 𝐿)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1342  wcel 2135  Vcvv 2722  cdif 3109  wss 3112  {csn 3571  cmpt 4038  ccnv 4598  cima 4602   Fn wfn 5178  wf 5179  cfv 5183  (class class class)co 5837  𝑓 cof 6043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-coll 4092  ax-sep 4095  ax-pow 4148  ax-pr 4182  ax-setind 4509
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-ral 2447  df-rex 2448  df-reu 2449  df-rab 2451  df-v 2724  df-sbc 2948  df-csb 3042  df-dif 3114  df-un 3116  df-in 3118  df-ss 3125  df-pw 3556  df-sn 3577  df-pr 3578  df-op 3580  df-uni 3785  df-iun 3863  df-br 3978  df-opab 4039  df-mpt 4040  df-id 4266  df-xp 4605  df-rel 4606  df-cnv 4607  df-co 4608  df-dm 4609  df-rn 4610  df-res 4611  df-ima 4612  df-iota 5148  df-fun 5185  df-fn 5186  df-f 5187  df-f1 5188  df-fo 5189  df-f1o 5190  df-fv 5191  df-ov 5840  df-oprab 5841  df-mpo 5842  df-of 6045
This theorem is referenced by: (None)
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