Theorem suppssof1 5670

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 4989	. . . . . 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 4989	. . . . . 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 3140	. . . . 5 ⊢ (𝐷 ∩ 𝐷) = 𝐷
9		eqidd 2038	. . . . 5 ⊢ ((φ ∧ x ∈ 𝐷) → (A‘x) = (A‘x))
10		eqidd 2038	. . . . 5 ⊢ ((φ ∧ x ∈ 𝐷) → (B‘x) = (B‘x))
11	3, 6, 7, 7, 8, 9, 10	offval 5661	. . . 4 ⊢ (φ → (A ∘𝑓 𝑂B) = (x ∈ 𝐷 ↦ ((A‘x)𝑂(B‘x))))
12	11	cnveqd 4454	. . 3 ⊢ (φ → ◡(A ∘𝑓 𝑂B) = ◡(x ∈ 𝐷 ↦ ((A‘x)𝑂(B‘x))))
13	12	imaeq1d 4610	. 2 ⊢ (φ → (◡(A ∘𝑓 𝑂B) “ (V ∖ {𝑍})) = (◡(x ∈ 𝐷 ↦ ((A‘x)𝑂(B‘x))) “ (V ∖ {𝑍})))
14	1	feqmptd 5169	. . . . . 6 ⊢ (φ → A = (x ∈ 𝐷 ↦ (A‘x)))
15	14	cnveqd 4454	. . . . 5 ⊢ (φ → ◡A = ◡(x ∈ 𝐷 ↦ (A‘x)))
16	15	imaeq1d 4610	. . . 4 ⊢ (φ → (◡A “ (V ∖ {𝑌})) = (◡(x ∈ 𝐷 ↦ (A‘x)) “ (V ∖ {𝑌})))
17		suppssof1.s	. . . 4 ⊢ (φ → (◡A “ (V ∖ {𝑌})) ⊆ 𝐿)
18	16, 17	eqsstr3d 2974	. . 3 ⊢ (φ → (◡(x ∈ 𝐷 ↦ (A‘x)) “ (V ∖ {𝑌})) ⊆ 𝐿)
19		suppssof1.o	. . 3 ⊢ ((φ ∧ v ∈ 𝑅) → (𝑌𝑂v) = 𝑍)
20		funfvex 5135	. . . . 5 ⊢ ((Fun A ∧ x ∈ dom A) → (A‘x) ∈ V)
21	20	funfni 4942	. . . 4 ⊢ ((A Fn 𝐷 ∧ x ∈ 𝐷) → (A‘x) ∈ V)
22	3, 21	sylan 267	. . 3 ⊢ ((φ ∧ x ∈ 𝐷) → (A‘x) ∈ V)
23	4	ffvelrnda 5245	. . 3 ⊢ ((φ ∧ x ∈ 𝐷) → (B‘x) ∈ 𝑅)
24	18, 19, 22, 23	suppssov1 5651	. 2 ⊢ (φ → (◡(x ∈ 𝐷 ↦ ((A‘x)𝑂(B‘x))) “ (V ∖ {𝑍})) ⊆ 𝐿)
25	13, 24	eqsstrd 2973	1 ⊢ (φ → (◡(A ∘𝑓 𝑂B) “ (V ∖ {𝑍})) ⊆ 𝐿)

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242   ∈ wcel 1390  Vcvv 2551   ∖ cdif 2908   ⊆ wss 2911  {csn 3367   ↦ cmpt 3809  ^◡ccnv 4287   “ cima 4291   Fn wfn 4840  ⟶wf 4841  ‘cfv 4845  (class class class)co 5455   ∘_𝑓 cof 5652

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-setind 4220

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-of 5654

This theorem is referenced by: (None)