Theorem suppssov1 6185

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 2791	. . . . . . . 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 3776	. . . . . . . 8 |- ( ( A O B ) e. ( _V \ { Z } ) -> ( A O B ) =/= Z )
6		oveq2 5982	. . . . . . . . . . . 12 |- ( v = B -> ( Y O v ) = ( Y O B ) )
7	6	eqeq1d 2218	. . . . . . . . . . 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 2583	. . . . . . . . . . . 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 2892	. . . . . . . . . 10 |- ( ( ph /\ x e. D ) -> ( Y O B ) = Z )
13		oveq1 5981	. . . . . . . . . . 11 |- ( A = Y -> ( A O B ) = ( Y O B ) )
14	13	eqeq1d 2218	. . . . . . . . . 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 2424	. . . . . . . 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 3774	. . . . . 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 3285	. . 3 |- ( ph -> { x e. D | ( A O B ) e. ( _V \ { Z } ) } C_ { x e. D | A e. ( _V \ { Y } ) } )
23		eqid 2209	. . . 4 |- ( x e. D |-> ( A O B ) ) = ( x e. D |-> ( A O B ) )
24	23	mptpreima 5198	. . 3 |- ( `' ( x e. D |-> ( A O B ) ) " ( _V \ { Z } ) ) = { x e. D | ( A O B ) e. ( _V \ { Z } ) }
25		eqid 2209	. . . 4 |- ( x e. D |-> A ) = ( x e. D |-> A )
26	25	mptpreima 5198	. . 3 |- ( `' ( x e. D |-> A ) " ( _V \ { Y } ) ) = { x e. D | A e. ( _V \ { Y } ) }
27	22, 24, 26	3sstr4g 3247	. 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 3214	1 |- ( ph -> ( `' ( x e. D |-> ( A O B ) ) " ( _V \ { Z } ) ) C_ L )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1375   e. wcel 2180   =/= wne 2380   A.wral 2488   {crab 2492   _Vcvv 2779   \ cdif 3174   C_ wss 3177   {csn 3646   |-> cmpt 4124   `'ccnv 4695   "cima 4699  (class class class)co 5974

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-ral 2493  df-rex 2494  df-rab 2497  df-v 2781  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-br 4063  df-opab 4125  df-mpt 4126  df-xp 4702  df-rel 4703  df-cnv 4704  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fv 5302  df-ov 5977

This theorem is referenced by:  suppssof1  6206