Theorem suppssov1 6132

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 2774	. . . . . . . 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 3751	. . . . . . . 8 |- ( ( A O B ) e. ( _V \ { Z } ) -> ( A O B ) =/= Z )
6		oveq2 5930	. . . . . . . . . . . 12 |- ( v = B -> ( Y O v ) = ( Y O B ) )
7	6	eqeq1d 2205	. . . . . . . . . . 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 2570	. . . . . . . . . . . 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 2873	. . . . . . . . . 10 |- ( ( ph /\ x e. D ) -> ( Y O B ) = Z )
13		oveq1 5929	. . . . . . . . . . 11 |- ( A = Y -> ( A O B ) = ( Y O B ) )
14	13	eqeq1d 2205	. . . . . . . . . 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 2411	. . . . . . . 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 3749	. . . . . 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 3264	. . 3 |- ( ph -> { x e. D | ( A O B ) e. ( _V \ { Z } ) } C_ { x e. D | A e. ( _V \ { Y } ) } )
23		eqid 2196	. . . 4 |- ( x e. D |-> ( A O B ) ) = ( x e. D |-> ( A O B ) )
24	23	mptpreima 5163	. . 3 |- ( `' ( x e. D |-> ( A O B ) ) " ( _V \ { Z } ) ) = { x e. D | ( A O B ) e. ( _V \ { Z } ) }
25		eqid 2196	. . . 4 |- ( x e. D |-> A ) = ( x e. D |-> A )
26	25	mptpreima 5163	. . 3 |- ( `' ( x e. D |-> A ) " ( _V \ { Y } ) ) = { x e. D | A e. ( _V \ { Y } ) }
27	22, 24, 26	3sstr4g 3226	. 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 3193	1 |- ( ph -> ( `' ( x e. D |-> ( A O B ) ) " ( _V \ { Z } ) ) C_ L )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   =/= wne 2367   A.wral 2475   {crab 2479   _Vcvv 2763   \ cdif 3154   C_ wss 3157   {csn 3622   |-> cmpt 4094   `'ccnv 4662   "cima 4666  (class class class)co 5922

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-xp 4669  df-rel 4670  df-cnv 4671  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fv 5266  df-ov 5925

This theorem is referenced by:  suppssof1  6153