Theorem suppssof1 6102

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 5367	. . . . . 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 5367	. . . . . 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 3346	. . . . 5 ⊢ (𝐷 ∩ 𝐷) = 𝐷
9		eqidd 2178	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) = (𝐴‘𝑥))
10		eqidd 2178	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐵‘𝑥) = (𝐵‘𝑥))
11	3, 6, 7, 7, 8, 9, 10	offval 6092	. . . 4 ⊢ (𝜑 → (𝐴 ∘𝑓 𝑂𝐵) = (𝑥 ∈ 𝐷 ↦ ((𝐴‘𝑥)𝑂(𝐵‘𝑥))))
12	11	cnveqd 4805	. . 3 ⊢ (𝜑 → ◡(𝐴 ∘𝑓 𝑂𝐵) = ◡(𝑥 ∈ 𝐷 ↦ ((𝐴‘𝑥)𝑂(𝐵‘𝑥))))
13	12	imaeq1d 4971	. 2 ⊢ (𝜑 → (◡(𝐴 ∘𝑓 𝑂𝐵) “ (V ∖ {𝑍})) = (◡(𝑥 ∈ 𝐷 ↦ ((𝐴‘𝑥)𝑂(𝐵‘𝑥))) “ (V ∖ {𝑍})))
14	1	feqmptd 5571	. . . . . 6 ⊢ (𝜑 → 𝐴 = (𝑥 ∈ 𝐷 ↦ (𝐴‘𝑥)))
15	14	cnveqd 4805	. . . . 5 ⊢ (𝜑 → ◡𝐴 = ◡(𝑥 ∈ 𝐷 ↦ (𝐴‘𝑥)))
16	15	imaeq1d 4971	. . . 4 ⊢ (𝜑 → (◡𝐴 “ (V ∖ {𝑌})) = (◡(𝑥 ∈ 𝐷 ↦ (𝐴‘𝑥)) “ (V ∖ {𝑌})))
17		suppssof1.s	. . . 4 ⊢ (𝜑 → (◡𝐴 “ (V ∖ {𝑌})) ⊆ 𝐿)
18	16, 17	eqsstrrd 3194	. . 3 ⊢ (𝜑 → (◡(𝑥 ∈ 𝐷 ↦ (𝐴‘𝑥)) “ (V ∖ {𝑌})) ⊆ 𝐿)
19		suppssof1.o	. . 3 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑅) → (𝑌𝑂𝑣) = 𝑍)
20		funfvex 5534	. . . . 5 ⊢ ((Fun 𝐴 ∧ 𝑥 ∈ dom 𝐴) → (𝐴‘𝑥) ∈ V)
21	20	funfni 5318	. . . 4 ⊢ ((𝐴 Fn 𝐷 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) ∈ V)
22	3, 21	sylan 283	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) ∈ V)
23	4	ffvelcdmda 5653	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐵‘𝑥) ∈ 𝑅)
24	18, 19, 22, 23	suppssov1 6082	. 2 ⊢ (𝜑 → (◡(𝑥 ∈ 𝐷 ↦ ((𝐴‘𝑥)𝑂(𝐵‘𝑥))) “ (V ∖ {𝑍})) ⊆ 𝐿)
25	13, 24	eqsstrd 3193	1 ⊢ (𝜑 → (◡(𝐴 ∘𝑓 𝑂𝐵) “ (V ∖ {𝑍})) ⊆ 𝐿)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1353   ∈ wcel 2148  Vcvv 2739   ∖ cdif 3128   ⊆ wss 3131  {csn 3594   ↦ cmpt 4066  ^◡ccnv 4627   “ cima 4631   Fn wfn 5213  ⟶wf 5214  ‘cfv 5218  (class class class)co 5877   ∘_𝑓 cof 6083

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-setind 4538

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-of 6085

This theorem is referenced by: (None)