Theorem suppssof1 5999

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 5272	. . . . . 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 5272	. . . . . 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 3285	. . . . 5 |- ( D i^i D ) = D
9		eqidd 2140	. . . . 5 |- ( ( ph /\ x e. D ) -> ( A ` x ) = ( A ` x ) )
10		eqidd 2140	. . . . 5 |- ( ( ph /\ x e. D ) -> ( B ` x ) = ( B ` x ) )
11	3, 6, 7, 7, 8, 9, 10	offval 5989	. . . 4 |- ( ph -> ( A oF O B ) = ( x e. D |-> ( ( A ` x ) O ( B ` x ) ) ) )
12	11	cnveqd 4715	. . 3 |- ( ph -> `' ( A oF O B ) = `' ( x e. D |-> ( ( A ` x ) O ( B ` x ) ) ) )
13	12	imaeq1d 4880	. 2 |- ( ph -> ( `' ( A oF O B ) " ( _V \ { Z } ) ) = ( `' ( x e. D |-> ( ( A ` x ) O ( B ` x ) ) ) " ( _V \ { Z } ) ) )
14	1	feqmptd 5474	. . . . . 6 |- ( ph -> A = ( x e. D |-> ( A ` x ) ) )
15	14	cnveqd 4715	. . . . 5 |- ( ph -> `' A = `' ( x e. D |-> ( A ` x ) ) )
16	15	imaeq1d 4880	. . . 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 3134	. . 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 5438	. . . . 5 |- ( ( Fun A /\ x e. dom A ) -> ( A ` x ) e. _V )
21	20	funfni 5223	. . . 4 |- ( ( A Fn D /\ x e. D ) -> ( A ` x ) e. _V )
22	3, 21	sylan 281	. . 3 |- ( ( ph /\ x e. D ) -> ( A ` x ) e. _V )
23	4	ffvelrnda 5555	. . 3 |- ( ( ph /\ x e. D ) -> ( B ` x ) e. R )
24	18, 19, 22, 23	suppssov1 5979	. 2 |- ( ph -> ( `' ( x e. D |-> ( ( A ` x ) O ( B ` x ) ) ) " ( _V \ { Z } ) ) C_ L )
25	13, 24	eqsstrd 3133	1 |- ( ph -> ( `' ( A oF O B ) " ( _V \ { Z } ) ) C_ L )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   _Vcvv 2686   \ cdif 3068   C_ wss 3071   {csn 3527   |-> cmpt 3989   `'ccnv 4538   "cima 4542   Fn wfn 5118   -->wf 5119   ` cfv 5123  (class class class)co 5774   oFcof 5980

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-setind 4452

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-of 5982

This theorem is referenced by: (None)