Theorem suppssof1 5647

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 4968	. . . . . 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 4968	. . . . . 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 3119	. . . . 5 ⊢ (𝐷 ∩ 𝐷) = 𝐷
9		eqidd 2019	. . . . 5 ⊢ ((φ ∧ x ∈ 𝐷) → (A‘x) = (A‘x))
10		eqidd 2019	. . . . 5 ⊢ ((φ ∧ x ∈ 𝐷) → (B‘x) = (B‘x))
11	3, 6, 7, 7, 8, 9, 10	offval 5638	. . . 4 ⊢ (φ → (A ∘𝑓 𝑂B) = (x ∈ 𝐷 ↦ ((A‘x)𝑂(B‘x))))
12	11	cnveqd 4434	. . 3 ⊢ (φ → ◡(A ∘𝑓 𝑂B) = ◡(x ∈ 𝐷 ↦ ((A‘x)𝑂(B‘x))))
13	12	imaeq1d 4590	. 2 ⊢ (φ → (◡(A ∘𝑓 𝑂B) “ (V ∖ {𝑍})) = (◡(x ∈ 𝐷 ↦ ((A‘x)𝑂(B‘x))) “ (V ∖ {𝑍})))
14	1	feqmptd 5147	. . . . . 6 ⊢ (φ → A = (x ∈ 𝐷 ↦ (A‘x)))
15	14	cnveqd 4434	. . . . 5 ⊢ (φ → ◡A = ◡(x ∈ 𝐷 ↦ (A‘x)))
16	15	imaeq1d 4590	. . . 4 ⊢ (φ → (◡A “ (V ∖ {𝑌})) = (◡(x ∈ 𝐷 ↦ (A‘x)) “ (V ∖ {𝑌})))
17		suppssof1.s	. . . 4 ⊢ (φ → (◡A “ (V ∖ {𝑌})) ⊆ 𝐿)
18	16, 17	eqsstr3d 2953	. . 3 ⊢ (φ → (◡(x ∈ 𝐷 ↦ (A‘x)) “ (V ∖ {𝑌})) ⊆ 𝐿)
19		suppssof1.o	. . 3 ⊢ ((φ ∧ v ∈ 𝑅) → (𝑌𝑂v) = 𝑍)
20		funfvex 5113	. . . . 5 ⊢ ((Fun A ∧ x ∈ dom A) → (A‘x) ∈ V)
21	20	funfni 4921	. . . 4 ⊢ ((A Fn 𝐷 ∧ x ∈ 𝐷) → (A‘x) ∈ V)
22	3, 21	sylan 267	. . 3 ⊢ ((φ ∧ x ∈ 𝐷) → (A‘x) ∈ V)
23	4	ffvelrnda 5223	. . 3 ⊢ ((φ ∧ x ∈ 𝐷) → (B‘x) ∈ 𝑅)
24	18, 19, 22, 23	suppssov1 5628	. 2 ⊢ (φ → (◡(x ∈ 𝐷 ↦ ((A‘x)𝑂(B‘x))) “ (V ∖ {𝑍})) ⊆ 𝐿)
25	13, 24	eqsstrd 2952	1 ⊢ (φ → (◡(A ∘𝑓 𝑂B) “ (V ∖ {𝑍})) ⊆ 𝐿)

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1226   ∈ wcel 1370  Vcvv 2531   ∖ cdif 2887   ⊆ wss 2890  {csn 3346   ↦ cmpt 3788  ^◡ccnv 4267   “ cima 4271   Fn wfn 4820  ⟶wf 4821  ‘cfv 4825  (class class class)co 5432   ∘_𝑓 cof 5629

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 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-coll 3842  ax-sep 3845  ax-pow 3897  ax-pr 3914  ax-setind 4200

This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-fal 1232  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ne 2184  df-ral 2285  df-rex 2286  df-reu 2287  df-rab 2289  df-v 2533  df-sbc 2738  df-csb 2826  df-dif 2893  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-iun 3629  df-br 3735  df-opab 3789  df-mpt 3790  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-iota 4790  df-fun 4827  df-fn 4828  df-f 4829  df-f1 4830  df-fo 4831  df-f1o 4832  df-fv 4833  df-ov 5435  df-oprab 5436  df-mpt2 5437  df-of 5631

This theorem is referenced by: (None)