Theorem suppssov1 5979

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 2697	. . . . . . . 8 |- ( A e. V -> A e. _V )
3	1, 2	syl 14	. . . . . . 7 |- ( ( ph /\ x e. D ) -> A e. _V )
4	3	adantr 274	. . . . . 6 |- ( ( ( ph /\ x e. D ) /\ ( A O B ) e. ( _V \ { Z } ) ) -> A e. _V )
5		eldifsni 3652	. . . . . . . 8 |- ( ( A O B ) e. ( _V \ { Z } ) -> ( A O B ) =/= Z )
6		oveq2 5782	. . . . . . . . . . . 12 |- ( v = B -> ( Y O v ) = ( Y O B ) )
7	6	eqeq1d 2148	. . . . . . . . . . 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 2505	. . . . . . . . . . . 12 |- ( ph -> A. v e. R ( Y O v ) = Z )
10	9	adantr 274	. . . . . . . . . . 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 2794	. . . . . . . . . 10 |- ( ( ph /\ x e. D ) -> ( Y O B ) = Z )
13		oveq1 5781	. . . . . . . . . . 11 |- ( A = Y -> ( A O B ) = ( Y O B ) )
14	13	eqeq1d 2148	. . . . . . . . . 10 |- ( A = Y -> ( ( A O B ) = Z <-> ( Y O B ) = Z ) )
15	12, 14	syl5ibrcom 156	. . . . . . . . 9 |- ( ( ph /\ x e. D ) -> ( A = Y -> ( A O B ) = Z ) )
16	15	necon3d 2352	. . . . . . . 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 123	. . . . . 6 |- ( ( ( ph /\ x e. D ) /\ ( A O B ) e. ( _V \ { Z } ) ) -> A =/= Y )
19		eldifsn 3650	. . . . . 6 |- ( A e. ( _V \ { Y } ) <-> ( A e. _V /\ A =/= Y ) )
20	4, 18, 19	sylanbrc 413	. . . . 5 |- ( ( ( ph /\ x e. D ) /\ ( A O B ) e. ( _V \ { Z } ) ) -> A e. ( _V \ { Y } ) )
21	20	ex 114	. . . 4 |- ( ( ph /\ x e. D ) -> ( ( A O B ) e. ( _V \ { Z } ) -> A e. ( _V \ { Y } ) ) )
22	21	ss2rabdv 3178	. . 3 |- ( ph -> { x e. D | ( A O B ) e. ( _V \ { Z } ) } C_ { x e. D | A e. ( _V \ { Y } ) } )
23		eqid 2139	. . . 4 |- ( x e. D |-> ( A O B ) ) = ( x e. D |-> ( A O B ) )
24	23	mptpreima 5032	. . 3 |- ( `' ( x e. D |-> ( A O B ) ) " ( _V \ { Z } ) ) = { x e. D | ( A O B ) e. ( _V \ { Z } ) }
25		eqid 2139	. . . 4 |- ( x e. D |-> A ) = ( x e. D |-> A )
26	25	mptpreima 5032	. . 3 |- ( `' ( x e. D |-> A ) " ( _V \ { Y } ) ) = { x e. D | A e. ( _V \ { Y } ) }
27	22, 24, 26	3sstr4g 3140	. 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 3107	1 |- ( ph -> ( `' ( x e. D |-> ( A O B ) ) " ( _V \ { Z } ) ) C_ L )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   =/= wne 2308   A.wral 2416   {crab 2420   _Vcvv 2686   \ cdif 3068   C_ wss 3071   {csn 3527   |-> cmpt 3989   `'ccnv 4538   "cima 4542  (class class class)co 5774

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-xp 4545  df-rel 4546  df-cnv 4547  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fv 5131  df-ov 5777

This theorem is referenced by:  suppssof1  5999