Theorem suppssof1 6078

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 5347	. . . . . 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 5347	. . . . . 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 3336	. . . . 5 |- ( D i^i D ) = D
9		eqidd 2171	. . . . 5 |- ( ( ph /\ x e. D ) -> ( A ` x ) = ( A ` x ) )
10		eqidd 2171	. . . . 5 |- ( ( ph /\ x e. D ) -> ( B ` x ) = ( B ` x ) )
11	3, 6, 7, 7, 8, 9, 10	offval 6068	. . . 4 |- ( ph -> ( A oF O B ) = ( x e. D |-> ( ( A ` x ) O ( B ` x ) ) ) )
12	11	cnveqd 4787	. . 3 |- ( ph -> `' ( A oF O B ) = `' ( x e. D |-> ( ( A ` x ) O ( B ` x ) ) ) )
13	12	imaeq1d 4952	. 2 |- ( ph -> ( `' ( A oF O B ) " ( _V \ { Z } ) ) = ( `' ( x e. D |-> ( ( A ` x ) O ( B ` x ) ) ) " ( _V \ { Z } ) ) )
14	1	feqmptd 5549	. . . . . 6 |- ( ph -> A = ( x e. D |-> ( A ` x ) ) )
15	14	cnveqd 4787	. . . . 5 |- ( ph -> `' A = `' ( x e. D |-> ( A ` x ) ) )
16	15	imaeq1d 4952	. . . 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 3184	. . 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 5513	. . . . 5 |- ( ( Fun A /\ x e. dom A ) -> ( A ` x ) e. _V )
21	20	funfni 5298	. . . 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 5631	. . 3 |- ( ( ph /\ x e. D ) -> ( B ` x ) e. R )
24	18, 19, 22, 23	suppssov1 6058	. 2 |- ( ph -> ( `' ( x e. D |-> ( ( A ` x ) O ( B ` x ) ) ) " ( _V \ { Z } ) ) C_ L )
25	13, 24	eqsstrd 3183	1 |- ( ph -> ( `' ( A oF O B ) " ( _V \ { Z } ) ) C_ L )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   _Vcvv 2730   \ cdif 3118   C_ wss 3121   {csn 3583   |-> cmpt 4050   `'ccnv 4610   "cima 4614   Fn wfn 5193   -->wf 5194   ` cfv 5198  (class class class)co 5853   oFcof 6059

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-of 6061

This theorem is referenced by: (None)