Theorem pimiooltgt 40921

Description: The preimage of an open interval is the intersection of the preimage of an unbounded below open interval and an unbounded above open interval. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
pimiooltgt.1	|- F/ x ph
pimiooltgt.2	|- ( ph -> L e. RR* )
pimiooltgt.3	|- ( ph -> R e. RR* )
pimiooltgt.4	|- ( ( ph /\ x e. A ) -> B e. RR* )

Assertion

Ref	Expression
pimiooltgt	|- ( ph -> { x e. A | B e. ( L (,) R ) } = ( { x e. A | B < R } i^i { x e. A | L < B } ) )

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   B( x)   R( x)   L( x)

Proof of Theorem pimiooltgt

Step	Hyp	Ref	Expression
1		pimiooltgt.1	. . . . 5 |- F/ x ph
2		pimiooltgt.2	. . . . . . . . 9 |- ( ph -> L e. RR* )
3	2	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> L e. RR* )
4	3	3adant3 1081	. . . . . . 7 |- ( ( ph /\ x e. A /\ B e. ( L (,) R ) ) -> L e. RR* )
5		pimiooltgt.3	. . . . . . . . 9 |- ( ph -> R e. RR* )
6	5	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> R e. RR* )
7	6	3adant3 1081	. . . . . . 7 |- ( ( ph /\ x e. A /\ B e. ( L (,) R ) ) -> R e. RR* )
8		simp3 1063	. . . . . . 7 |- ( ( ph /\ x e. A /\ B e. ( L (,) R ) ) -> B e. ( L (,) R ) )
9		iooltub 39735	. . . . . . 7 |- ( ( L e. RR* /\ R e. RR* /\ B e. ( L (,) R ) ) -> B < R )
10	4, 7, 8, 9	syl3anc 1326	. . . . . 6 |- ( ( ph /\ x e. A /\ B e. ( L (,) R ) ) -> B < R )
11	10	3exp 1264	. . . . 5 |- ( ph -> ( x e. A -> ( B e. ( L (,) R ) -> B < R ) ) )
12	1, 11	ralrimi 2957	. . . 4 |- ( ph -> A. x e. A ( B e. ( L (,) R ) -> B < R ) )
13		ss2rab 3678	. . . 4 |- ( { x e. A | B e. ( L (,) R ) } C_ { x e. A | B < R } <-> A. x e. A ( B e. ( L (,) R ) -> B < R ) )
14	12, 13	sylibr 224	. . 3 |- ( ph -> { x e. A | B e. ( L (,) R ) } C_ { x e. A | B < R } )
15		ioogtlb 39717	. . . . . . 7 |- ( ( L e. RR* /\ R e. RR* /\ B e. ( L (,) R ) ) -> L < B )
16	4, 7, 8, 15	syl3anc 1326	. . . . . 6 |- ( ( ph /\ x e. A /\ B e. ( L (,) R ) ) -> L < B )
17	16	3exp 1264	. . . . 5 |- ( ph -> ( x e. A -> ( B e. ( L (,) R ) -> L < B ) ) )
18	1, 17	ralrimi 2957	. . . 4 |- ( ph -> A. x e. A ( B e. ( L (,) R ) -> L < B ) )
19		ss2rab 3678	. . . 4 |- ( { x e. A | B e. ( L (,) R ) } C_ { x e. A | L < B } <-> A. x e. A ( B e. ( L (,) R ) -> L < B ) )
20	18, 19	sylibr 224	. . 3 |- ( ph -> { x e. A | B e. ( L (,) R ) } C_ { x e. A | L < B } )
21	14, 20	ssind 3837	. 2 |- ( ph -> { x e. A | B e. ( L (,) R ) } C_ ( { x e. A | B < R } i^i { x e. A | L < B } ) )
22		elinel1 3799	. . . . . . . . 9 |- ( x e. ( { x e. A | B < R } i^i { x e. A | L < B } ) -> x e. { x e. A | B < R } )
23		ssrab2 3687	. . . . . . . . . 10 |- { x e. A | B < R } C_ A
24	23	sseli 3599	. . . . . . . . 9 |- ( x e. { x e. A | B < R } -> x e. A )
25	22, 24	syl 17	. . . . . . . 8 |- ( x e. ( { x e. A | B < R } i^i { x e. A | L < B } ) -> x e. A )
26	25	adantl 482	. . . . . . 7 |- ( ( ph /\ x e. ( { x e. A | B < R } i^i { x e. A | L < B } ) ) -> x e. A )
27	26, 3	syldan 487	. . . . . . . 8 |- ( ( ph /\ x e. ( { x e. A | B < R } i^i { x e. A | L < B } ) ) -> L e. RR* )
28	26, 6	syldan 487	. . . . . . . 8 |- ( ( ph /\ x e. ( { x e. A | B < R } i^i { x e. A | L < B } ) ) -> R e. RR* )
29		pimiooltgt.4	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> B e. RR* )
30	26, 29	syldan 487	. . . . . . . . 9 |- ( ( ph /\ x e. ( { x e. A | B < R } i^i { x e. A | L < B } ) ) -> B e. RR* )
31		mnfxr 10096	. . . . . . . . . . . 12 |- -oo e. RR*
32	31	a1i 11	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( { x e. A | B < R } i^i { x e. A | L < B } ) ) -> -oo e. RR* )
33	27	mnfled 39609	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( { x e. A | B < R } i^i { x e. A | L < B } ) ) -> -oo <_ L )
34		elinel2 3800	. . . . . . . . . . . . . 14 |- ( x e. ( { x e. A | B < R } i^i { x e. A | L < B } ) -> x e. { x e. A | L < B } )
35		rabidim2 39284	. . . . . . . . . . . . . 14 |- ( x e. { x e. A | L < B } -> L < B )
36	34, 35	syl 17	. . . . . . . . . . . . 13 |- ( x e. ( { x e. A | B < R } i^i { x e. A | L < B } ) -> L < B )
37	36	adantl 482	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( { x e. A | B < R } i^i { x e. A | L < B } ) ) -> L < B )
38	32, 27, 30, 33, 37	xrlelttrd 11991	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( { x e. A | B < R } i^i { x e. A | L < B } ) ) -> -oo < B )
39	32, 30, 38	xrltned 39573	. . . . . . . . . 10 |- ( ( ph /\ x e. ( { x e. A | B < R } i^i { x e. A | L < B } ) ) -> -oo =/= B )
40	39	necomd 2849	. . . . . . . . 9 |- ( ( ph /\ x e. ( { x e. A | B < R } i^i { x e. A | L < B } ) ) -> B =/= -oo )
41		pnfxr 10092	. . . . . . . . . . 11 |- +oo e. RR*
42	41	a1i 11	. . . . . . . . . 10 |- ( ( ph /\ x e. ( { x e. A | B < R } i^i { x e. A | L < B } ) ) -> +oo e. RR* )
43		rabidim2 39284	. . . . . . . . . . . . 13 |- ( x e. { x e. A | B < R } -> B < R )
44	22, 43	syl 17	. . . . . . . . . . . 12 |- ( x e. ( { x e. A | B < R } i^i { x e. A | L < B } ) -> B < R )
45	44	adantl 482	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( { x e. A | B < R } i^i { x e. A | L < B } ) ) -> B < R )
46		pnfge 11964	. . . . . . . . . . . 12 |- ( R e. RR* -> R <_ +oo )
47	28, 46	syl 17	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( { x e. A | B < R } i^i { x e. A | L < B } ) ) -> R <_ +oo )
48	30, 28, 42, 45, 47	xrltletrd 11992	. . . . . . . . . 10 |- ( ( ph /\ x e. ( { x e. A | B < R } i^i { x e. A | L < B } ) ) -> B < +oo )
49	30, 42, 48	xrltned 39573	. . . . . . . . 9 |- ( ( ph /\ x e. ( { x e. A | B < R } i^i { x e. A | L < B } ) ) -> B =/= +oo )
50	30, 40, 49	xrred 39581	. . . . . . . 8 |- ( ( ph /\ x e. ( { x e. A | B < R } i^i { x e. A | L < B } ) ) -> B e. RR )
51	27, 28, 50, 37, 45	eliood 39720	. . . . . . 7 |- ( ( ph /\ x e. ( { x e. A | B < R } i^i { x e. A | L < B } ) ) -> B e. ( L (,) R ) )
52	26, 51	jca 554	. . . . . 6 |- ( ( ph /\ x e. ( { x e. A | B < R } i^i { x e. A | L < B } ) ) -> ( x e. A /\ B e. ( L (,) R ) ) )
53		rabid 3116	. . . . . 6 |- ( x e. { x e. A | B e. ( L (,) R ) } <-> ( x e. A /\ B e. ( L (,) R ) ) )
54	52, 53	sylibr 224	. . . . 5 |- ( ( ph /\ x e. ( { x e. A | B < R } i^i { x e. A | L < B } ) ) -> x e. { x e. A | B e. ( L (,) R ) } )
55	54	ex 450	. . . 4 |- ( ph -> ( x e. ( { x e. A | B < R } i^i { x e. A | L < B } ) -> x e. { x e. A | B e. ( L (,) R ) } ) )
56	1, 55	ralrimi 2957	. . 3 |- ( ph -> A. x e. ( { x e. A | B < R } i^i { x e. A | L < B } ) x e. { x e. A | B e. ( L (,) R ) } )
57		nfrab1 3122	. . . . 5 |- F/_ x { x e. A | B < R }
58		nfrab1 3122	. . . . 5 |- F/_ x { x e. A | L < B }
59	57, 58	nfin 3820	. . . 4 |- F/_ x ( { x e. A | B < R } i^i { x e. A | L < B } )
60		nfrab1 3122	. . . 4 |- F/_ x { x e. A | B e. ( L (,) R ) }
61	59, 60	dfss3f 3595	. . 3 |- ( ( { x e. A | B < R } i^i { x e. A | L < B } ) C_ { x e. A | B e. ( L (,) R ) } <-> A. x e. ( { x e. A | B < R } i^i { x e. A | L < B } ) x e. { x e. A | B e. ( L (,) R ) } )
62	56, 61	sylibr 224	. 2 |- ( ph -> ( { x e. A | B < R } i^i { x e. A | L < B } ) C_ { x e. A | B e. ( L (,) R ) } )
63	21, 62	eqssd 3620	1 |- ( ph -> { x e. A | B e. ( L (,) R ) } = ( { x e. A | B < R } i^i { x e. A | L < B } ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   F/wnf 1708   e. wcel 1990   A.wral 2912   {crab 2916   i^i cin 3573   C_ wss 3574   class class class wbr 4653  (class class class)co 6650   +oocpnf 10071   -oocmnf 10072   RR*cxr 10073   < clt 10074   <_ cle 10075   (,)cioo 12175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-ioo 12179

This theorem is referenced by:  smfpimioompt  40993