Theorem preimaiocmnf 39788

Description: Preimage of a right-closed interval, unbounded below. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
preimaiocmnf.1	|- ( ph -> F : A --> RR )
preimaiocmnf.2	|- ( ph -> B e. RR* )

Assertion

Ref	Expression
preimaiocmnf	|- ( ph -> ( `' F " ( -oo (,] B ) ) = { x e. A | ( F ` x ) <_ B } )

Distinct variable groups:   x, A   x, B   x, F   ph, x

Proof of Theorem preimaiocmnf

Step	Hyp	Ref	Expression
1		preimaiocmnf.1	. . . 4 |- ( ph -> F : A --> RR )
2	1	ffnd 6046	. . 3 |- ( ph -> F Fn A )
3		fncnvima2 6339	. . 3 |- ( F Fn A -> ( `' F " ( -oo (,] B ) ) = { x e. A | ( F ` x ) e. ( -oo (,] B ) } )
4	2, 3	syl 17	. 2 |- ( ph -> ( `' F " ( -oo (,] B ) ) = { x e. A | ( F ` x ) e. ( -oo (,] B ) } )
5		mnfxr 10096	. . . . . . . 8 |- -oo e. RR*
6	5	a1i 11	. . . . . . 7 |- ( ( ph /\ ( F ` x ) e. ( -oo (,] B ) ) -> -oo e. RR* )
7		preimaiocmnf.2	. . . . . . . 8 |- ( ph -> B e. RR* )
8	7	adantr 481	. . . . . . 7 |- ( ( ph /\ ( F ` x ) e. ( -oo (,] B ) ) -> B e. RR* )
9		simpr 477	. . . . . . 7 |- ( ( ph /\ ( F ` x ) e. ( -oo (,] B ) ) -> ( F ` x ) e. ( -oo (,] B ) )
10	6, 8, 9	iocleubd 39786	. . . . . 6 |- ( ( ph /\ ( F ` x ) e. ( -oo (,] B ) ) -> ( F ` x ) <_ B )
11	10	ex 450	. . . . 5 |- ( ph -> ( ( F ` x ) e. ( -oo (,] B ) -> ( F ` x ) <_ B ) )
12	11	adantr 481	. . . 4 |- ( ( ph /\ x e. A ) -> ( ( F ` x ) e. ( -oo (,] B ) -> ( F ` x ) <_ B ) )
13	5	a1i 11	. . . . . 6 |- ( ( ( ph /\ x e. A ) /\ ( F ` x ) <_ B ) -> -oo e. RR* )
14	7	adantr 481	. . . . . . 7 |- ( ( ph /\ ( F ` x ) <_ B ) -> B e. RR* )
15	14	adantlr 751	. . . . . 6 |- ( ( ( ph /\ x e. A ) /\ ( F ` x ) <_ B ) -> B e. RR* )
16	1	ffvelrnda 6359	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> ( F ` x ) e. RR )
17	16	rexrd 10089	. . . . . . 7 |- ( ( ph /\ x e. A ) -> ( F ` x ) e. RR* )
18	17	adantr 481	. . . . . 6 |- ( ( ( ph /\ x e. A ) /\ ( F ` x ) <_ B ) -> ( F ` x ) e. RR* )
19	16	mnfltd 11958	. . . . . . 7 |- ( ( ph /\ x e. A ) -> -oo < ( F ` x ) )
20	19	adantr 481	. . . . . 6 |- ( ( ( ph /\ x e. A ) /\ ( F ` x ) <_ B ) -> -oo < ( F ` x ) )
21		simpr 477	. . . . . 6 |- ( ( ( ph /\ x e. A ) /\ ( F ` x ) <_ B ) -> ( F ` x ) <_ B )
22	13, 15, 18, 20, 21	eliocd 39730	. . . . 5 |- ( ( ( ph /\ x e. A ) /\ ( F ` x ) <_ B ) -> ( F ` x ) e. ( -oo (,] B ) )
23	22	ex 450	. . . 4 |- ( ( ph /\ x e. A ) -> ( ( F ` x ) <_ B -> ( F ` x ) e. ( -oo (,] B ) ) )
24	12, 23	impbid 202	. . 3 |- ( ( ph /\ x e. A ) -> ( ( F ` x ) e. ( -oo (,] B ) <-> ( F ` x ) <_ B ) )
25	24	rabbidva 3188	. 2 |- ( ph -> { x e. A | ( F ` x ) e. ( -oo (,] B ) } = { x e. A | ( F ` x ) <_ B } )
26	4, 25	eqtrd 2656	1 |- ( ph -> ( `' F " ( -oo (,] B ) ) = { x e. A | ( F ` x ) <_ B } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   class class class wbr 4653   `'ccnv 5113   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-id 5024  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-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-ioc 12180

This theorem is referenced by:  issmfle2d  41015