Theorem suppssov1 5709

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 2566	. . . . . . . 8 |- ( A e. V -> A e. _V )
3	1, 2	syl 14	. . . . . . 7 |- ( ( ph /\ x e. D ) -> A e. _V )
4	3	adantr 261	. . . . . 6 |- ( ( ( ph /\ x e. D ) /\ ( A O B ) e. ( _V \ { Z } ) ) -> A e. _V )
5		eldifsni 3496	. . . . . . . 8 |- ( ( A O B ) e. ( _V \ { Z } ) -> ( A O B ) =/= Z )
6		suppssov1.b	. . . . . . . . . . 11 |- ( ( ph /\ x e. D ) -> B e. R )
7		suppssov1.o	. . . . . . . . . . . . 13 |- ( ( ph /\ v e. R ) -> ( Y O v ) = Z )
8	7	ralrimiva 2392	. . . . . . . . . . . 12 |- ( ph -> A. v e. R ( Y O v ) = Z )
9	8	adantr 261	. . . . . . . . . . 11 |- ( ( ph /\ x e. D ) -> A. v e. R ( Y O v ) = Z )
10		oveq2 5520	. . . . . . . . . . . . 13 |- ( v = B -> ( Y O v ) = ( Y O B ) )
11	10	eqeq1d 2048	. . . . . . . . . . . 12 |- ( v = B -> ( ( Y O v ) = Z <-> ( Y O B ) = Z ) )
12	11	rspcva 2654	. . . . . . . . . . 11 |- ( ( B e. R /\ A. v e. R ( Y O v ) = Z ) -> ( Y O B ) = Z )
13	6, 9, 12	syl2anc 391	. . . . . . . . . 10 |- ( ( ph /\ x e. D ) -> ( Y O B ) = Z )
14		oveq1 5519	. . . . . . . . . . 11 |- ( A = Y -> ( A O B ) = ( Y O B ) )
15	14	eqeq1d 2048	. . . . . . . . . 10 |- ( A = Y -> ( ( A O B ) = Z <-> ( Y O B ) = Z ) )
16	13, 15	syl5ibrcom 146	. . . . . . . . 9 |- ( ( ph /\ x e. D ) -> ( A = Y -> ( A O B ) = Z ) )
17	16	necon3d 2249	. . . . . . . 8 |- ( ( ph /\ x e. D ) -> ( ( A O B ) =/= Z -> A =/= Y ) )
18	5, 17	syl5 28	. . . . . . 7 |- ( ( ph /\ x e. D ) -> ( ( A O B ) e. ( _V \ { Z } ) -> A =/= Y ) )
19	18	imp 115	. . . . . 6 |- ( ( ( ph /\ x e. D ) /\ ( A O B ) e. ( _V \ { Z } ) ) -> A =/= Y )
20		eldifsn 3495	. . . . . 6 |- ( A e. ( _V \ { Y } ) <-> ( A e. _V /\ A =/= Y ) )
21	4, 19, 20	sylanbrc 394	. . . . 5 |- ( ( ( ph /\ x e. D ) /\ ( A O B ) e. ( _V \ { Z } ) ) -> A e. ( _V \ { Y } ) )
22	21	ex 108	. . . 4 |- ( ( ph /\ x e. D ) -> ( ( A O B ) e. ( _V \ { Z } ) -> A e. ( _V \ { Y } ) ) )
23	22	ss2rabdv 3021	. . 3 |- ( ph -> { x e. D | ( A O B ) e. ( _V \ { Z } ) } C_ { x e. D | A e. ( _V \ { Y } ) } )
24		eqid 2040	. . . 4 |- ( x e. D |-> ( A O B ) ) = ( x e. D |-> ( A O B ) )
25	24	mptpreima 4814	. . 3 |- ( `' ( x e. D |-> ( A O B ) ) " ( _V \ { Z } ) ) = { x e. D | ( A O B ) e. ( _V \ { Z } ) }
26		eqid 2040	. . . 4 |- ( x e. D |-> A ) = ( x e. D |-> A )
27	26	mptpreima 4814	. . 3 |- ( `' ( x e. D |-> A ) " ( _V \ { Y } ) ) = { x e. D | A e. ( _V \ { Y } ) }
28	23, 25, 27	3sstr4g 2986	. 2 |- ( ph -> ( `' ( x e. D |-> ( A O B ) ) " ( _V \ { Z } ) ) C_ ( `' ( x e. D |-> A ) " ( _V \ { Y } ) ) )
29		suppssov1.s	. 2 |- ( ph -> ( `' ( x e. D |-> A ) " ( _V \ { Y } ) ) C_ L )
30	28, 29	sstrd 2955	1 |- ( ph -> ( `' ( x e. D |-> ( A O B ) ) " ( _V \ { Z } ) ) C_ L )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   e. wcel 1393   =/= wne 2204   A.wral 2306   {crab 2310   _Vcvv 2557   \ cdif 2914   C_ wss 2917   {csn 3375   |-> cmpt 3818   `'ccnv 4344   "cima 4348  (class class class)co 5512

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-xp 4351  df-rel 4352  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fv 4910  df-ov 5515

This theorem is referenced by:  suppssof1  5728