Theorem suppssov1OLD 6532

Description: Formula building theorem for support restrictions: operator with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.) Obsolete version of suppssov1 6951 as of 28-May-2019. (New usage is discouraged.)

Hypotheses

Ref	Expression
suppssov1OLD.s
suppssov1OLD.o
suppssov1OLD.a
suppssov1OLD.b

Assertion

Ref	Expression
suppssov1OLD

Distinct variable groups:   ,   ,   ,   ,O   ,   ,   ,   ,   ,

Proof of Theorem suppssov1OLD

Step	Hyp	Ref	Expression
1		suppssov1OLD.a	. . . . . . . 8
2		elex 3118	. . . . . . . 8
3	1, 2	syl 16	. . . . . . 7
4	3	adantr 465	. . . . . 6
5		eldifsni 4156	. . . . . . . 8
6		suppssov1OLD.b	. . . . . . . . . . 11
7		suppssov1OLD.o	. . . . . . . . . . . . 13
8	7	ralrimiva 2871	. . . . . . . . . . . 12
9	8	adantr 465	. . . . . . . . . . 11
10		oveq2 6304	. . . . . . . . . . . . 13
11	10	eqeq1d 2459	. . . . . . . . . . . 12
12	11	rspcva 3208	. . . . . . . . . . 11
13	6, 9, 12	syl2anc 661	. . . . . . . . . 10
14		oveq1 6303	. . . . . . . . . . 11
15	14	eqeq1d 2459	. . . . . . . . . 10
16	13, 15	syl5ibrcom 222	. . . . . . . . 9
17	16	necon3d 2681	. . . . . . . 8
18	5, 17	syl5 32	. . . . . . 7
19	18	imp 429	. . . . . 6
20		eldifsn 4155	. . . . . 6
21	4, 19, 20	sylanbrc 664	. . . . 5
22	21	ex 434	. . . 4
23	22	ss2rabdv 3580	. . 3
24		eqid 2457	. . . 4
25	24	mptpreima 5505	. . 3
26		eqid 2457	. . . 4
27	26	mptpreima 5505	. . 3
28	23, 25, 27	3sstr4g 3544	. 2
29		suppssov1OLD.s	. 2
30	28, 29	sstrd 3513	1

Syntax hints:  ->wi 4  /\wa 369  =wceq 1395  e.wcel 1818  =/=wne 2652  A.wral 2807  {crab 2811  cvv 3109  \cdif 3472  C_wss 3475  {csn 4029  e.cmpt 4510  `'ccnv 5003  "cima 5007  (class class class)co 6296

This theorem is referenced by:  suppssof1OLD  6559  evlslem6OLD  18182

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-mpt 4512  df-xp 5010  df-rel 5011  df-cnv 5012  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fv 5601  df-ov 6299