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	|- (ph -> ( `'A "( _V \ { Y })) C_ L)
suppssof1.o	|- ((ph /\ v e. R) -> ( Y Ov) = Z)
suppssof1.a	|- (ph -> A : D --> V)
suppssof1.b	|- (ph -> B : D --> R)
suppssof1.d	|- (ph -> D e. W)

Assertion

Ref	Expression
suppssof1	|- (ph -> ( `'(A o F OB) "( _V \ { Z })) C_ L)

Distinct variable groups:   ph,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 |- (ph -> A : D --> V)
2		ffn 4989	. . . . . 6 |- (A : D --> V -> A Fn D)
3	1, 2	syl 14	. . . . 5 |- (ph -> A Fn D)
4		suppssof1.b	. . . . . 6 |- (ph -> B : D --> R)
5		ffn 4989	. . . . . 6 |- (B : D --> R -> B Fn D)
6	4, 5	syl 14	. . . . 5 |- (ph -> B Fn D)
7		suppssof1.d	. . . . 5 |- (ph -> D e. W)
8		inidm 3140	. . . . 5 |- ( D i^i D) = D
9		eqidd 2038	. . . . 5 |- ((ph /\ x e. D) -> (A ` x) = (A ` x))
10		eqidd 2038	. . . . 5 |- ((ph /\ x e. D) -> (B ` x) = (B ` x))
11	3, 6, 7, 7, 8, 9, 10	offval 5661	. . . 4 |- (ph -> (A o F OB) = (x e. D |-> ((A ` x) O(B ` x))))
12	11	cnveqd 4454	. . 3 |- (ph -> `'(A o F OB) = `'(x e. D |-> ((A ` x) O(B ` x))))
13	12	imaeq1d 4610	. 2 |- (ph -> ( `'(A o F OB) "( _V \ { Z })) = ( `'(x e. D |-> ((A ` x) O(B ` x))) "( _V \ { Z })))
14	1	feqmptd 5169	. . . . . 6 |- (ph -> A = (x e. D |-> (A ` x)))
15	14	cnveqd 4454	. . . . 5 |- (ph -> `'A = `'(x e. D |-> (A ` x)))
16	15	imaeq1d 4610	. . . 4 |- (ph -> ( `'A "( _V \ { Y })) = ( `'(x e. D |-> (A ` x)) "( _V \ { Y })))
17		suppssof1.s	. . . 4 |- (ph -> ( `'A "( _V \ { Y })) C_ L)
18	16, 17	eqsstr3d 2974	. . 3 |- (ph -> ( `'(x e. D |-> (A ` x)) "( _V \ { Y })) C_ L)
19		suppssof1.o	. . 3 |- ((ph /\ v e. R) -> ( Y Ov) = Z)
20		funfvex 5135	. . . . 5 |- (( Fun A /\ x e. dom A) -> (A ` x) e. _V)
21	20	funfni 4942	. . . 4 |- ((A Fn D /\ x e. D) -> (A ` x) e. _V)
22	3, 21	sylan 267	. . 3 |- ((ph /\ x e. D) -> (A ` x) e. _V)
23	4	ffvelrnda 5245	. . 3 |- ((ph /\ x e. D) -> (B ` x) e. R)
24	18, 19, 22, 23	suppssov1 5651	. 2 |- (ph -> ( `'(x e. D |-> ((A ` x) O(B ` x))) "( _V \ { Z })) C_ L)
25	13, 24	eqsstrd 2973	1 |- (ph -> ( `'(A o F OB) "( _V \ { Z })) C_ L)

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1242   e. wcel 1390   _Vcvv 2551   \ cdif 2908   C_ wss 2911   {csn 3367   |-> cmpt 3809   `'ccnv 4287   "cima 4291   Fn wfn 4840   -->wf 4841   ` cfv 4845  (class class class)co 5455   o Fcof 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-bndl 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)