Theorem suppssof1 6067

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 5337	. . . . . 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 5337	. . . . . 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 3331	. . . . 5 |- ( D i^i D ) = D
9		eqidd 2166	. . . . 5 |- ( ( ph /\ x e. D ) -> ( A ` x ) = ( A ` x ) )
10		eqidd 2166	. . . . 5 |- ( ( ph /\ x e. D ) -> ( B ` x ) = ( B ` x ) )
11	3, 6, 7, 7, 8, 9, 10	offval 6057	. . . 4 |- ( ph -> ( A oF O B ) = ( x e. D |-> ( ( A ` x ) O ( B ` x ) ) ) )
12	11	cnveqd 4780	. . 3 |- ( ph -> `' ( A oF O B ) = `' ( x e. D |-> ( ( A ` x ) O ( B ` x ) ) ) )
13	12	imaeq1d 4945	. 2 |- ( ph -> ( `' ( A oF O B ) " ( _V \ { Z } ) ) = ( `' ( x e. D |-> ( ( A ` x ) O ( B ` x ) ) ) " ( _V \ { Z } ) ) )
14	1	feqmptd 5539	. . . . . 6 |- ( ph -> A = ( x e. D |-> ( A ` x ) ) )
15	14	cnveqd 4780	. . . . 5 |- ( ph -> `' A = `' ( x e. D |-> ( A ` x ) ) )
16	15	imaeq1d 4945	. . . 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 3179	. . 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 5503	. . . . 5 |- ( ( Fun A /\ x e. dom A ) -> ( A ` x ) e. _V )
21	20	funfni 5288	. . . 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 5620	. . 3 |- ( ( ph /\ x e. D ) -> ( B ` x ) e. R )
24	18, 19, 22, 23	suppssov1 6047	. 2 |- ( ph -> ( `' ( x e. D |-> ( ( A ` x ) O ( B ` x ) ) ) " ( _V \ { Z } ) ) C_ L )
25	13, 24	eqsstrd 3178	1 |- ( ph -> ( `' ( A oF O B ) " ( _V \ { Z } ) ) C_ L )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   _Vcvv 2726   \ cdif 3113   C_ wss 3116   {csn 3576   |-> cmpt 4043   `'ccnv 4603   "cima 4607   Fn wfn 5183   -->wf 5184   ` cfv 5188  (class class class)co 5842   oFcof 6048

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-of 6050

This theorem is referenced by: (None)