Theorem liminfval 39991

Description: The inferior limit of a set F. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypothesis

Ref	Expression
liminfval.1	|- G = ( k e. RR |-> inf ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) )

Assertion

Ref	Expression
liminfval	|- ( F e. V -> (liminf ` F ) = sup ( ran G , RR* , < ) )

Distinct variable group:   k, F

Allowed substitution hints:   G( k)   V( k)

Proof of Theorem liminfval

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-liminf 39984	. 2 |- liminf = ( x e. _V |-> sup ( ran ( k e. RR |-> inf ( ( ( x " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) ) , RR* , < ) )
2		imaeq1 5461	. . . . . . . . 9 |- ( x = F -> ( x " ( k [,) +oo ) ) = ( F " ( k [,) +oo ) ) )
3	2	ineq1d 3813	. . . . . . . 8 |- ( x = F -> ( ( x " ( k [,) +oo ) ) i^i RR* ) = ( ( F " ( k [,) +oo ) ) i^i RR* ) )
4	3	infeq1d 8383	. . . . . . 7 |- ( x = F -> inf ( ( ( x " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) = inf ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) )
5	4	mpteq2dv 4745	. . . . . 6 |- ( x = F -> ( k e. RR |-> inf ( ( ( x " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) ) = ( k e. RR |-> inf ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) ) )
6		liminfval.1	. . . . . . 7 |- G = ( k e. RR |-> inf ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) )
7	6	a1i 11	. . . . . 6 |- ( x = F -> G = ( k e. RR |-> inf ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) ) )
8	5, 7	eqtr4d 2659	. . . . 5 |- ( x = F -> ( k e. RR |-> inf ( ( ( x " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) ) = G )
9	8	rneqd 5353	. . . 4 |- ( x = F -> ran ( k e. RR |-> inf ( ( ( x " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) ) = ran G )
10	9	supeq1d 8352	. . 3 |- ( x = F -> sup ( ran ( k e. RR |-> inf ( ( ( x " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) ) , RR* , < ) = sup ( ran G , RR* , < ) )
11	10	adantl 482	. 2 |- ( ( F e. V /\ x = F ) -> sup ( ran ( k e. RR |-> inf ( ( ( x " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) ) , RR* , < ) = sup ( ran G , RR* , < ) )
12		elex 3212	. 2 |- ( F e. V -> F e. _V )
13		xrltso 11974	. . . 4 |- < Or RR*
14	13	supex 8369	. . 3 |- sup ( ran G , RR* , < ) e. _V
15	14	a1i 11	. 2 |- ( F e. V -> sup ( ran G , RR* , < ) e. _V )
16	1, 11, 12, 15	fvmptd2 39445	1 |- ( F e. V -> (liminf ` F ) = sup ( ran G , RR* , < ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   |-> cmpt 4729   ran crn 5115   "cima 5117   ` cfv 5888  (class class class)co 6650   supcsup 8346  infcinf 8347   RRcr 9935   +oocpnf 10071   RR*cxr 10073   < clt 10074   [,)cico 12177  liminfclsi 39983

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-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-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-liminf 39984

This theorem is referenced by:  liminfcl  39995  liminfvald  39996  liminfval5  39997  liminfresxr  39999  liminfval2  40000  liminfvalxr  40015