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Theorem suppssof1 6124
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 5384 . . . . . 6 (𝐴:𝐷𝑉𝐴 Fn 𝐷)
31, 2syl 14 . . . . 5 (𝜑𝐴 Fn 𝐷)
4 suppssof1.b . . . . . 6 (𝜑𝐵:𝐷𝑅)
5 ffn 5384 . . . . . 6 (𝐵:𝐷𝑅𝐵 Fn 𝐷)
64, 5syl 14 . . . . 5 (𝜑𝐵 Fn 𝐷)
7 suppssof1.d . . . . 5 (𝜑𝐷𝑊)
8 inidm 3359 . . . . 5 (𝐷𝐷) = 𝐷
9 eqidd 2190 . . . . 5 ((𝜑𝑥𝐷) → (𝐴𝑥) = (𝐴𝑥))
10 eqidd 2190 . . . . 5 ((𝜑𝑥𝐷) → (𝐵𝑥) = (𝐵𝑥))
113, 6, 7, 7, 8, 9, 10offval 6114 . . . 4 (𝜑 → (𝐴𝑓 𝑂𝐵) = (𝑥𝐷 ↦ ((𝐴𝑥)𝑂(𝐵𝑥))))
1211cnveqd 4821 . . 3 (𝜑(𝐴𝑓 𝑂𝐵) = (𝑥𝐷 ↦ ((𝐴𝑥)𝑂(𝐵𝑥))))
1312imaeq1d 4987 . 2 (𝜑 → ((𝐴𝑓 𝑂𝐵) “ (V ∖ {𝑍})) = ((𝑥𝐷 ↦ ((𝐴𝑥)𝑂(𝐵𝑥))) “ (V ∖ {𝑍})))
141feqmptd 5590 . . . . . 6 (𝜑𝐴 = (𝑥𝐷 ↦ (𝐴𝑥)))
1514cnveqd 4821 . . . . 5 (𝜑𝐴 = (𝑥𝐷 ↦ (𝐴𝑥)))
1615imaeq1d 4987 . . . 4 (𝜑 → (𝐴 “ (V ∖ {𝑌})) = ((𝑥𝐷 ↦ (𝐴𝑥)) “ (V ∖ {𝑌})))
17 suppssof1.s . . . 4 (𝜑 → (𝐴 “ (V ∖ {𝑌})) ⊆ 𝐿)
1816, 17eqsstrrd 3207 . . 3 (𝜑 → ((𝑥𝐷 ↦ (𝐴𝑥)) “ (V ∖ {𝑌})) ⊆ 𝐿)
19 suppssof1.o . . 3 ((𝜑𝑣𝑅) → (𝑌𝑂𝑣) = 𝑍)
20 funfvex 5551 . . . . 5 ((Fun 𝐴𝑥 ∈ dom 𝐴) → (𝐴𝑥) ∈ V)
2120funfni 5335 . . . 4 ((𝐴 Fn 𝐷𝑥𝐷) → (𝐴𝑥) ∈ V)
223, 21sylan 283 . . 3 ((𝜑𝑥𝐷) → (𝐴𝑥) ∈ V)
234ffvelcdmda 5672 . . 3 ((𝜑𝑥𝐷) → (𝐵𝑥) ∈ 𝑅)
2418, 19, 22, 23suppssov1 6103 . 2 (𝜑 → ((𝑥𝐷 ↦ ((𝐴𝑥)𝑂(𝐵𝑥))) “ (V ∖ {𝑍})) ⊆ 𝐿)
2513, 24eqsstrd 3206 1 (𝜑 → ((𝐴𝑓 𝑂𝐵) “ (V ∖ {𝑍})) ⊆ 𝐿)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wcel 2160  Vcvv 2752  cdif 3141  wss 3144  {csn 3607  cmpt 4079  ccnv 4643  cima 4647   Fn wfn 5230  wf 5231  cfv 5235  (class class class)co 5896  𝑓 cof 6104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-setind 4554
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5899  df-oprab 5900  df-mpo 5901  df-of 6106
This theorem is referenced by: (None)
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