Theorem pimltmnf2 40911

Description: Given a real-valued function, the preimage of an open interval, unbounded below, with upper bound -oo, is the empty set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
pimltmnf2.1	|- F/_ x F
pimltmnf2.2	|- ( ph -> F : A --> RR )

Assertion

Ref	Expression
pimltmnf2	|- ( ph -> { x e. A | ( F ` x ) < -oo } = (/) )

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   F( x)

Proof of Theorem pimltmnf2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfcv 2764	. . . 4 |- F/_ x A
2		nfcv 2764	. . . 4 |- F/_ y A
3		nfv 1843	. . . 4 |- F/ y ( F ` x ) < -oo
4		pimltmnf2.1	. . . . . 6 |- F/_ x F
5		nfcv 2764	. . . . . 6 |- F/_ x y
6	4, 5	nffv 6198	. . . . 5 |- F/_ x ( F ` y )
7		nfcv 2764	. . . . 5 |- F/_ x <
8		nfcv 2764	. . . . 5 |- F/_ x -oo
9	6, 7, 8	nfbr 4699	. . . 4 |- F/ x ( F ` y ) < -oo
10		fveq2 6191	. . . . 5 |- ( x = y -> ( F ` x ) = ( F ` y ) )
11	10	breq1d 4663	. . . 4 |- ( x = y -> ( ( F ` x ) < -oo <-> ( F ` y ) < -oo ) )
12	1, 2, 3, 9, 11	cbvrab 3198	. . 3 |- { x e. A | ( F ` x ) < -oo } = { y e. A | ( F ` y ) < -oo }
13	12	a1i 11	. 2 |- ( ph -> { x e. A | ( F ` x ) < -oo } = { y e. A | ( F ` y ) < -oo } )
14		mnfxr 10096	. . . . . . 7 |- -oo e. RR*
15	14	a1i 11	. . . . . 6 |- ( ( ph /\ y e. A ) -> -oo e. RR* )
16		pimltmnf2.2	. . . . . . . 8 |- ( ph -> F : A --> RR )
17	16	ffvelrnda 6359	. . . . . . 7 |- ( ( ph /\ y e. A ) -> ( F ` y ) e. RR )
18	17	rexrd 10089	. . . . . 6 |- ( ( ph /\ y e. A ) -> ( F ` y ) e. RR* )
19	17	mnfltd 11958	. . . . . 6 |- ( ( ph /\ y e. A ) -> -oo < ( F ` y ) )
20	15, 18, 19	xrltled 39486	. . . . 5 |- ( ( ph /\ y e. A ) -> -oo <_ ( F ` y ) )
21	15, 18	xrlenltd 10104	. . . . 5 |- ( ( ph /\ y e. A ) -> ( -oo <_ ( F ` y ) <-> -. ( F ` y ) < -oo ) )
22	20, 21	mpbid 222	. . . 4 |- ( ( ph /\ y e. A ) -> -. ( F ` y ) < -oo )
23	22	ralrimiva 2966	. . 3 |- ( ph -> A. y e. A -. ( F ` y ) < -oo )
24		rabeq0 3957	. . 3 |- ( { y e. A | ( F ` y ) < -oo } = (/) <-> A. y e. A -. ( F ` y ) < -oo )
25	23, 24	sylibr 224	. 2 |- ( ph -> { y e. A | ( F ` y ) < -oo } = (/) )
26	13, 25	eqtrd 2656	1 |- ( ph -> { x e. A | ( F ` x ) < -oo } = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   F/_wnfc 2751   A.wral 2912   {crab 2916   (/)c0 3915   class class class wbr 4653   -->wf 5884   ` cfv 5888   RRcr 9935   -oocmnf 10072   RR*cxr 10073   < clt 10074   <_ cle 10075

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-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-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

This theorem is referenced by:  smfpimltxr  40956