Theorem suppssof1 6314

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 O v ) = 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 oF O B ) " ( _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 5531	. . . . . 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 5531	. . . . . 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 3440	. . . . 5 |- ( D i^i D ) = D
9		eqidd 2239	. . . . 5 |- ( ( ph /\ x e. D ) -> ( A ` x ) = ( A ` x ) )
10		eqidd 2239	. . . . 5 |- ( ( ph /\ x e. D ) -> ( B ` x ) = ( B ` x ) )
11	3, 6, 7, 7, 8, 9, 10	offval 6304	. . . 4 |- ( ph -> ( A oF O B ) = ( x e. D |-> ( ( A ` x ) O ( B ` x ) ) ) )
12	11	cnveqd 4954	. . 3 |- ( ph -> `' ( A oF O B ) = `' ( x e. D |-> ( ( A ` x ) O ( B ` x ) ) ) )
13	12	imaeq1d 5123	. 2 |- ( ph -> ( `' ( A oF O B ) " ( _V \ { Z } ) ) = ( `' ( x e. D |-> ( ( A ` x ) O ( B ` x ) ) ) " ( _V \ { Z } ) ) )
14	1	feqmptd 5753	. . . . . 6 |- ( ph -> A = ( x e. D |-> ( A ` x ) ) )
15	14	cnveqd 4954	. . . . 5 |- ( ph -> `' A = `' ( x e. D |-> ( A ` x ) ) )
16	15	imaeq1d 5123	. . . 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	eqsstrrd 3285	. . 3 |- ( ph -> ( `' ( x e. D |-> ( A ` x ) ) " ( _V \ { Y } ) ) C_ L )
19		suppssof1.o	. . 3 |- ( ( ph /\ v e. R ) -> ( Y O v ) = Z )
20		funfvex 5710	. . . . 5 |- ( ( Fun A /\ x e. dom A ) -> ( A ` x ) e. _V )
21	20	funfni 5481	. . . 4 |- ( ( A Fn D /\ x e. D ) -> ( A ` x ) e. _V )
22	3, 21	sylan 283	. . 3 |- ( ( ph /\ x e. D ) -> ( A ` x ) e. _V )
23	4	ffvelcdmda 5837	. . 3 |- ( ( ph /\ x e. D ) -> ( B ` x ) e. R )
24	18, 19, 22, 23	suppssov1 6293	. 2 |- ( ph -> ( `' ( x e. D |-> ( ( A ` x ) O ( B ` x ) ) ) " ( _V \ { Z } ) ) C_ L )
25	13, 24	eqsstrd 3284	1 |- ( ph -> ( `' ( A oF O B ) " ( _V \ { Z } ) ) C_ L )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   _Vcvv 2821   \ cdif 3217   C_ wss 3220   {csn 3708   |-> cmpt 4190   `'ccnv 4771   "cima 4775   Fn wfn 5370   -->wf 5371   ` cfv 5375  (class class class)co 6079   oFcof 6294

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-setind 4682

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-of 6296

This theorem is referenced by: (None)