Theorem suppssof1 6007

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 5280	. . . . . 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 5280	. . . . . 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 3290	. . . . 5 |- ( D i^i D ) = D
9		eqidd 2141	. . . . 5 |- ( ( ph /\ x e. D ) -> ( A ` x ) = ( A ` x ) )
10		eqidd 2141	. . . . 5 |- ( ( ph /\ x e. D ) -> ( B ` x ) = ( B ` x ) )
11	3, 6, 7, 7, 8, 9, 10	offval 5997	. . . 4 |- ( ph -> ( A oF O B ) = ( x e. D |-> ( ( A ` x ) O ( B ` x ) ) ) )
12	11	cnveqd 4723	. . 3 |- ( ph -> `' ( A oF O B ) = `' ( x e. D |-> ( ( A ` x ) O ( B ` x ) ) ) )
13	12	imaeq1d 4888	. 2 |- ( ph -> ( `' ( A oF O B ) " ( _V \ { Z } ) ) = ( `' ( x e. D |-> ( ( A ` x ) O ( B ` x ) ) ) " ( _V \ { Z } ) ) )
14	1	feqmptd 5482	. . . . . 6 |- ( ph -> A = ( x e. D |-> ( A ` x ) ) )
15	14	cnveqd 4723	. . . . 5 |- ( ph -> `' A = `' ( x e. D |-> ( A ` x ) ) )
16	15	imaeq1d 4888	. . . 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 3139	. . 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 5446	. . . . 5 |- ( ( Fun A /\ x e. dom A ) -> ( A ` x ) e. _V )
21	20	funfni 5231	. . . 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 5563	. . 3 |- ( ( ph /\ x e. D ) -> ( B ` x ) e. R )
24	18, 19, 22, 23	suppssov1 5987	. 2 |- ( ph -> ( `' ( x e. D |-> ( ( A ` x ) O ( B ` x ) ) ) " ( _V \ { Z } ) ) C_ L )
25	13, 24	eqsstrd 3138	1 |- ( ph -> ( `' ( A oF O B ) " ( _V \ { Z } ) ) C_ L )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1332   e. wcel 1481   _Vcvv 2689   \ cdif 3073   C_ wss 3076   {csn 3532   |-> cmpt 3997   `'ccnv 4546   "cima 4550   Fn wfn 5126   -->wf 5127   ` cfv 5131  (class class class)co 5782   oFcof 5988

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-setind 4460

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-of 5990

This theorem is referenced by: (None)