Theorem suppssov1 6227

Description: Formula building theorem for support restrictions: operator with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.)

Hypotheses

Ref	Expression
suppssov1.s	|- ( ph -> ( `' ( x e. D |-> A ) " ( _V \ { Y } ) ) C_ L )
suppssov1.o	|- ( ( ph /\ v e. R ) -> ( Y O v ) = Z )
suppssov1.a	|- ( ( ph /\ x e. D ) -> A e. V )
suppssov1.b	|- ( ( ph /\ x e. D ) -> B e. R )

Assertion

Ref	Expression
suppssov1	|- ( ph -> ( `' ( x e. D |-> ( A O B ) ) " ( _V \ { Z } ) ) C_ L )

Distinct variable groups:   ph, v   ph, x   v, B   v, O   v, R   v, Y   x, Y   v, Z   x, Z

Allowed substitution hints:   A( x, v)   B( x)   D( x, v)   R( x)   L( x, v)   O( x)   V( x, v)

Proof of Theorem suppssov1

Step	Hyp	Ref	Expression
1		suppssov1.a	. . . . . . . 8 |- ( ( ph /\ x e. D ) -> A e. V )
2		elex 2812	. . . . . . . 8 |- ( A e. V -> A e. _V )
3	1, 2	syl 14	. . . . . . 7 |- ( ( ph /\ x e. D ) -> A e. _V )
4	3	adantr 276	. . . . . 6 |- ( ( ( ph /\ x e. D ) /\ ( A O B ) e. ( _V \ { Z } ) ) -> A e. _V )
5		eldifsni 3800	. . . . . . . 8 |- ( ( A O B ) e. ( _V \ { Z } ) -> ( A O B ) =/= Z )
6		oveq2 6021	. . . . . . . . . . . 12 |- ( v = B -> ( Y O v ) = ( Y O B ) )
7	6	eqeq1d 2238	. . . . . . . . . . 11 |- ( v = B -> ( ( Y O v ) = Z <-> ( Y O B ) = Z ) )
8		suppssov1.o	. . . . . . . . . . . . 13 |- ( ( ph /\ v e. R ) -> ( Y O v ) = Z )
9	8	ralrimiva 2603	. . . . . . . . . . . 12 |- ( ph -> A. v e. R ( Y O v ) = Z )
10	9	adantr 276	. . . . . . . . . . 11 |- ( ( ph /\ x e. D ) -> A. v e. R ( Y O v ) = Z )
11		suppssov1.b	. . . . . . . . . . 11 |- ( ( ph /\ x e. D ) -> B e. R )
12	7, 10, 11	rspcdva 2913	. . . . . . . . . 10 |- ( ( ph /\ x e. D ) -> ( Y O B ) = Z )
13		oveq1 6020	. . . . . . . . . . 11 |- ( A = Y -> ( A O B ) = ( Y O B ) )
14	13	eqeq1d 2238	. . . . . . . . . 10 |- ( A = Y -> ( ( A O B ) = Z <-> ( Y O B ) = Z ) )
15	12, 14	syl5ibrcom 157	. . . . . . . . 9 |- ( ( ph /\ x e. D ) -> ( A = Y -> ( A O B ) = Z ) )
16	15	necon3d 2444	. . . . . . . 8 |- ( ( ph /\ x e. D ) -> ( ( A O B ) =/= Z -> A =/= Y ) )
17	5, 16	syl5 32	. . . . . . 7 |- ( ( ph /\ x e. D ) -> ( ( A O B ) e. ( _V \ { Z } ) -> A =/= Y ) )
18	17	imp 124	. . . . . 6 |- ( ( ( ph /\ x e. D ) /\ ( A O B ) e. ( _V \ { Z } ) ) -> A =/= Y )
19		eldifsn 3798	. . . . . 6 |- ( A e. ( _V \ { Y } ) <-> ( A e. _V /\ A =/= Y ) )
20	4, 18, 19	sylanbrc 417	. . . . 5 |- ( ( ( ph /\ x e. D ) /\ ( A O B ) e. ( _V \ { Z } ) ) -> A e. ( _V \ { Y } ) )
21	20	ex 115	. . . 4 |- ( ( ph /\ x e. D ) -> ( ( A O B ) e. ( _V \ { Z } ) -> A e. ( _V \ { Y } ) ) )
22	21	ss2rabdv 3306	. . 3 |- ( ph -> { x e. D | ( A O B ) e. ( _V \ { Z } ) } C_ { x e. D | A e. ( _V \ { Y } ) } )
23		eqid 2229	. . . 4 |- ( x e. D |-> ( A O B ) ) = ( x e. D |-> ( A O B ) )
24	23	mptpreima 5228	. . 3 |- ( `' ( x e. D |-> ( A O B ) ) " ( _V \ { Z } ) ) = { x e. D | ( A O B ) e. ( _V \ { Z } ) }
25		eqid 2229	. . . 4 |- ( x e. D |-> A ) = ( x e. D |-> A )
26	25	mptpreima 5228	. . 3 |- ( `' ( x e. D |-> A ) " ( _V \ { Y } ) ) = { x e. D | A e. ( _V \ { Y } ) }
27	22, 24, 26	3sstr4g 3268	. 2 |- ( ph -> ( `' ( x e. D |-> ( A O B ) ) " ( _V \ { Z } ) ) C_ ( `' ( x e. D |-> A ) " ( _V \ { Y } ) ) )
28		suppssov1.s	. 2 |- ( ph -> ( `' ( x e. D |-> A ) " ( _V \ { Y } ) ) C_ L )
29	27, 28	sstrd 3235	1 |- ( ph -> ( `' ( x e. D |-> ( A O B ) ) " ( _V \ { Z } ) ) C_ L )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   =/= wne 2400   A.wral 2508   {crab 2512   _Vcvv 2800   \ cdif 3195   C_ wss 3198   {csn 3667   |-> cmpt 4148   `'ccnv 4722   "cima 4726  (class class class)co 6013

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-mpt 4150  df-xp 4729  df-rel 4730  df-cnv 4731  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fv 5332  df-ov 6016

This theorem is referenced by:  suppssof1  6248