Theorem pimincfltioc 40926

Description: Given a non decreasing function, the preimage of an unbounded below, open interval, when the supremum of the preimage belongs to the preimage. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
pimincfltioc.x	|- F/ x ph
pimincfltioc.h	|- F/ y ph
pimincfltioc.a	|- ( ph -> A C_ RR )
pimincfltioc.f	|- ( ph -> F : A --> RR* )
pimincfltioc.i	|- ( ph -> A. x e. A A. y e. A ( x <_ y -> ( F ` x ) <_ ( F ` y ) ) )
pimincfltioc.r	|- ( ph -> R e. RR* )
pimincfltioc.y	|- Y = { x e. A | ( F ` x ) < R }
pimincfltioc.c	|- S = sup ( Y , RR* , < )
pimincfltioc.e	|- ( ph -> S e. Y )
pimincfltioc.d	|- I = ( -oo (,] S )

Assertion

Ref	Expression
pimincfltioc	|- ( ph -> Y = ( I i^i A ) )

Distinct variable groups:   x, A, y   x, F, y   x, I, y   x, R   x, S, y

Allowed substitution hints:   ph( x, y)   R( y)   Y( x, y)

Proof of Theorem pimincfltioc

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pimincfltioc.y	. . . . . . 7 |- Y = { x e. A | ( F ` x ) < R }
2		ssrab2 3687	. . . . . . 7 |- { x e. A | ( F ` x ) < R } C_ A
3	1, 2	eqsstri 3635	. . . . . 6 |- Y C_ A
4	3	a1i 11	. . . . 5 |- ( ph -> Y C_ A )
5		pimincfltioc.a	. . . . 5 |- ( ph -> A C_ RR )
6	4, 5	sstrd 3613	. . . 4 |- ( ph -> Y C_ RR )
7		pimincfltioc.c	. . . 4 |- S = sup ( Y , RR* , < )
8		pimincfltioc.e	. . . 4 |- ( ph -> S e. Y )
9		pimincfltioc.d	. . . 4 |- I = ( -oo (,] S )
10	6, 7, 8, 9	ressiocsup 39781	. . 3 |- ( ph -> Y C_ I )
11	10, 4	ssind 3837	. 2 |- ( ph -> Y C_ ( I i^i A ) )
12		pimincfltioc.x	. . . 4 |- F/ x ph
13		elinel2 3800	. . . . . . . 8 |- ( x e. ( I i^i A ) -> x e. A )
14	13	adantl 482	. . . . . . 7 |- ( ( ph /\ x e. ( I i^i A ) ) -> x e. A )
15		pimincfltioc.f	. . . . . . . . . 10 |- ( ph -> F : A --> RR* )
16	15	adantr 481	. . . . . . . . 9 |- ( ( ph /\ x e. ( I i^i A ) ) -> F : A --> RR* )
17	16, 14	ffvelrnd 6360	. . . . . . . 8 |- ( ( ph /\ x e. ( I i^i A ) ) -> ( F ` x ) e. RR* )
18	3, 8	sseldi 3601	. . . . . . . . . 10 |- ( ph -> S e. A )
19	15, 18	ffvelrnd 6360	. . . . . . . . 9 |- ( ph -> ( F ` S ) e. RR* )
20	19	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. ( I i^i A ) ) -> ( F ` S ) e. RR* )
21		pimincfltioc.r	. . . . . . . . 9 |- ( ph -> R e. RR* )
22	21	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. ( I i^i A ) ) -> R e. RR* )
23		eleq1 2689	. . . . . . . . . . 11 |- ( z = x -> ( z e. ( I i^i A ) <-> x e. ( I i^i A ) ) )
24	23	anbi2d 740	. . . . . . . . . 10 |- ( z = x -> ( ( ph /\ z e. ( I i^i A ) ) <-> ( ph /\ x e. ( I i^i A ) ) ) )
25		fveq2 6191	. . . . . . . . . . 11 |- ( z = x -> ( F ` z ) = ( F ` x ) )
26	25	breq1d 4663	. . . . . . . . . 10 |- ( z = x -> ( ( F ` z ) <_ ( F ` S ) <-> ( F ` x ) <_ ( F ` S ) ) )
27	24, 26	imbi12d 334	. . . . . . . . 9 |- ( z = x -> ( ( ( ph /\ z e. ( I i^i A ) ) -> ( F ` z ) <_ ( F ` S ) ) <-> ( ( ph /\ x e. ( I i^i A ) ) -> ( F ` x ) <_ ( F ` S ) ) ) )
28		nfv 1843	. . . . . . . . . . 11 |- F/ x z e. ( I i^i A )
29	12, 28	nfan 1828	. . . . . . . . . 10 |- F/ x ( ph /\ z e. ( I i^i A ) )
30		pimincfltioc.h	. . . . . . . . . . 11 |- F/ y ph
31		nfv 1843	. . . . . . . . . . 11 |- F/ y z e. ( I i^i A )
32	30, 31	nfan 1828	. . . . . . . . . 10 |- F/ y ( ph /\ z e. ( I i^i A ) )
33		pimincfltioc.i	. . . . . . . . . . 11 |- ( ph -> A. x e. A A. y e. A ( x <_ y -> ( F ` x ) <_ ( F ` y ) ) )
34	33	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ z e. ( I i^i A ) ) -> A. x e. A A. y e. A ( x <_ y -> ( F ` x ) <_ ( F ` y ) ) )
35		elinel2 3800	. . . . . . . . . . 11 |- ( z e. ( I i^i A ) -> z e. A )
36	35	adantl 482	. . . . . . . . . 10 |- ( ( ph /\ z e. ( I i^i A ) ) -> z e. A )
37	18	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ z e. ( I i^i A ) ) -> S e. A )
38		mnfxr 10096	. . . . . . . . . . . 12 |- -oo e. RR*
39	38	a1i 11	. . . . . . . . . . 11 |- ( ( ph /\ z e. ( I i^i A ) ) -> -oo e. RR* )
40		ressxr 10083	. . . . . . . . . . . . 13 |- RR C_ RR*
41	6, 8	sseldd 3604	. . . . . . . . . . . . 13 |- ( ph -> S e. RR )
42	40, 41	sseldi 3601	. . . . . . . . . . . 12 |- ( ph -> S e. RR* )
43	42	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ z e. ( I i^i A ) ) -> S e. RR* )
44		elinel1 3799	. . . . . . . . . . . . 13 |- ( z e. ( I i^i A ) -> z e. I )
45	44, 9	syl6eleq 2711	. . . . . . . . . . . 12 |- ( z e. ( I i^i A ) -> z e. ( -oo (,] S ) )
46	45	adantl 482	. . . . . . . . . . 11 |- ( ( ph /\ z e. ( I i^i A ) ) -> z e. ( -oo (,] S ) )
47		iocleub 39725	. . . . . . . . . . 11 |- ( ( -oo e. RR* /\ S e. RR* /\ z e. ( -oo (,] S ) ) -> z <_ S )
48	39, 43, 46, 47	syl3anc 1326	. . . . . . . . . 10 |- ( ( ph /\ z e. ( I i^i A ) ) -> z <_ S )
49	29, 32, 34, 36, 37, 48	dmrelrnrel 39419	. . . . . . . . 9 |- ( ( ph /\ z e. ( I i^i A ) ) -> ( F ` z ) <_ ( F ` S ) )
50	27, 49	chvarv 2263	. . . . . . . 8 |- ( ( ph /\ x e. ( I i^i A ) ) -> ( F ` x ) <_ ( F ` S ) )
51		fveq2 6191	. . . . . . . . . . . . 13 |- ( x = S -> ( F ` x ) = ( F ` S ) )
52	51	breq1d 4663	. . . . . . . . . . . 12 |- ( x = S -> ( ( F ` x ) < R <-> ( F ` S ) < R ) )
53	52, 1	elrab2 3366	. . . . . . . . . . 11 |- ( S e. Y <-> ( S e. A /\ ( F ` S ) < R ) )
54	8, 53	sylib 208	. . . . . . . . . 10 |- ( ph -> ( S e. A /\ ( F ` S ) < R ) )
55	54	simprd 479	. . . . . . . . 9 |- ( ph -> ( F ` S ) < R )
56	55	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. ( I i^i A ) ) -> ( F ` S ) < R )
57	17, 20, 22, 50, 56	xrlelttrd 11991	. . . . . . 7 |- ( ( ph /\ x e. ( I i^i A ) ) -> ( F ` x ) < R )
58	14, 57	jca 554	. . . . . 6 |- ( ( ph /\ x e. ( I i^i A ) ) -> ( x e. A /\ ( F ` x ) < R ) )
59	1	rabeq2i 3197	. . . . . 6 |- ( x e. Y <-> ( x e. A /\ ( F ` x ) < R ) )
60	58, 59	sylibr 224	. . . . 5 |- ( ( ph /\ x e. ( I i^i A ) ) -> x e. Y )
61	60	ex 450	. . . 4 |- ( ph -> ( x e. ( I i^i A ) -> x e. Y ) )
62	12, 61	ralrimi 2957	. . 3 |- ( ph -> A. x e. ( I i^i A ) x e. Y )
63	28	nfci 2754	. . . 4 |- F/_ x ( I i^i A )
64		nfrab1 3122	. . . . 5 |- F/_ x { x e. A | ( F ` x ) < R }
65	1, 64	nfcxfr 2762	. . . 4 |- F/_ x Y
66	63, 65	dfss3f 3595	. . 3 |- ( ( I i^i A ) C_ Y <-> A. x e. ( I i^i A ) x e. Y )
67	62, 66	sylibr 224	. 2 |- ( ph -> ( I i^i A ) C_ Y )
68	11, 67	eqssd 3620	1 |- ( ph -> Y = ( I i^i A ) )

Syntax hints:   -> wi 4   /\ wa 384   = 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   -->wf 5884   ` cfv 5888  (class class class)co 6650   supcsup 8346   RRcr 9935   -oocmnf 10072   RR*cxr 10073   < clt 10074   <_ cle 10075   (,]cioc 12176

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-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

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-reu 2919  df-rmo 2920  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-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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-ioc 12180

This theorem is referenced by:  incsmflem  40950