Theorem suppssof1 6150

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	⊢ (𝜑 → 𝐴:𝐷⟶𝑉)
suppssof1.b	⊢ (𝜑 → 𝐵:𝐷⟶𝑅)
suppssof1.d	⊢ (𝜑 → 𝐷 ∈ 𝑊)

Assertion

Ref	Expression
suppssof1	⊢ (𝜑 → (◡(𝐴 ∘𝑓 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.

Step	Hyp	Ref	Expression
1		suppssof1.a	. . . . . 6 ⊢ (𝜑 → 𝐴:𝐷⟶𝑉)
2		ffn 5404	. . . . . 6 ⊢ (𝐴:𝐷⟶𝑉 → 𝐴 Fn 𝐷)
3	1, 2	syl 14	. . . . 5 ⊢ (𝜑 → 𝐴 Fn 𝐷)
4		suppssof1.b	. . . . . 6 ⊢ (𝜑 → 𝐵:𝐷⟶𝑅)
5		ffn 5404	. . . . . 6 ⊢ (𝐵:𝐷⟶𝑅 → 𝐵 Fn 𝐷)
6	4, 5	syl 14	. . . . 5 ⊢ (𝜑 → 𝐵 Fn 𝐷)
7		suppssof1.d	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑊)
8		inidm 3369	. . . . 5 ⊢ (𝐷 ∩ 𝐷) = 𝐷
9		eqidd 2194	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) = (𝐴‘𝑥))
10		eqidd 2194	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐵‘𝑥) = (𝐵‘𝑥))
11	3, 6, 7, 7, 8, 9, 10	offval 6140	. . . 4 ⊢ (𝜑 → (𝐴 ∘𝑓 𝑂𝐵) = (𝑥 ∈ 𝐷 ↦ ((𝐴‘𝑥)𝑂(𝐵‘𝑥))))
12	11	cnveqd 4839	. . 3 ⊢ (𝜑 → ◡(𝐴 ∘𝑓 𝑂𝐵) = ◡(𝑥 ∈ 𝐷 ↦ ((𝐴‘𝑥)𝑂(𝐵‘𝑥))))
13	12	imaeq1d 5005	. 2 ⊢ (𝜑 → (◡(𝐴 ∘𝑓 𝑂𝐵) “ (V ∖ {𝑍})) = (◡(𝑥 ∈ 𝐷 ↦ ((𝐴‘𝑥)𝑂(𝐵‘𝑥))) “ (V ∖ {𝑍})))
14	1	feqmptd 5611	. . . . . 6 ⊢ (𝜑 → 𝐴 = (𝑥 ∈ 𝐷 ↦ (𝐴‘𝑥)))
15	14	cnveqd 4839	. . . . 5 ⊢ (𝜑 → ◡𝐴 = ◡(𝑥 ∈ 𝐷 ↦ (𝐴‘𝑥)))
16	15	imaeq1d 5005	. . . 4 ⊢ (𝜑 → (◡𝐴 “ (V ∖ {𝑌})) = (◡(𝑥 ∈ 𝐷 ↦ (𝐴‘𝑥)) “ (V ∖ {𝑌})))
17		suppssof1.s	. . . 4 ⊢ (𝜑 → (◡𝐴 “ (V ∖ {𝑌})) ⊆ 𝐿)
18	16, 17	eqsstrrd 3217	. . 3 ⊢ (𝜑 → (◡(𝑥 ∈ 𝐷 ↦ (𝐴‘𝑥)) “ (V ∖ {𝑌})) ⊆ 𝐿)
19		suppssof1.o	. . 3 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑅) → (𝑌𝑂𝑣) = 𝑍)
20		funfvex 5572	. . . . 5 ⊢ ((Fun 𝐴 ∧ 𝑥 ∈ dom 𝐴) → (𝐴‘𝑥) ∈ V)
21	20	funfni 5355	. . . 4 ⊢ ((𝐴 Fn 𝐷 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) ∈ V)
22	3, 21	sylan 283	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) ∈ V)
23	4	ffvelcdmda 5694	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐵‘𝑥) ∈ 𝑅)
24	18, 19, 22, 23	suppssov1 6129	. 2 ⊢ (𝜑 → (◡(𝑥 ∈ 𝐷 ↦ ((𝐴‘𝑥)𝑂(𝐵‘𝑥))) “ (V ∖ {𝑍})) ⊆ 𝐿)
25	13, 24	eqsstrd 3216	1 ⊢ (𝜑 → (◡(𝐴 ∘𝑓 𝑂𝐵) “ (V ∖ {𝑍})) ⊆ 𝐿)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2164  Vcvv 2760   ∖ cdif 3151   ⊆ wss 3154  {csn 3619   ↦ cmpt 4091  ^◡ccnv 4659   “ cima 4663   Fn wfn 5250  ⟶wf 5251  ‘cfv 5255  (class class class)co 5919   ∘_𝑓 cof 6130

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 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-setind 4570

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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-of 6132

This theorem is referenced by: (None)