Theorem allbutfifvre 39907

Description: Given a sequence of real-valued functions, and X that belongs to all but finitely many domains, then its function value is ultimately a real number. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
allbutfifvre.1	|- F/ m ph
allbutfifvre.2	|- Z = ( ZZ>= ` M )
allbutfifvre.3	|- ( ( ph /\ m e. Z ) -> ( F ` m ) : dom ( F ` m ) --> RR )
allbutfifvre.4	|- D = U_ n e. Z |^|_ m e. ( ZZ>= ` n ) dom ( F ` m )
allbutfifvre.5	|- ( ph -> X e. D )

Assertion

Ref	Expression
allbutfifvre	|- ( ph -> E. n e. Z A. m e. ( ZZ>= ` n ) ( ( F ` m ) ` X ) e. RR )

Distinct variable groups:   m, X, n   m, Z   ph, n

Allowed substitution hints:   ph( m)   D( m, n)   F( m, n)   M( m, n)   Z( n)

Proof of Theorem allbutfifvre

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		allbutfifvre.5	. . . 4 |- ( ph -> X e. D )
2		allbutfifvre.4	. . . 4 |- D = U_ n e. Z |^|_ m e. ( ZZ>= ` n ) dom ( F ` m )
3	1, 2	syl6eleq 2711	. . 3 |- ( ph -> X e. U_ n e. Z |^|_ m e. ( ZZ>= ` n ) dom ( F ` m ) )
4		allbutfifvre.2	. . . 4 |- Z = ( ZZ>= ` M )
5		eqid 2622	. . . 4 |- U_ n e. Z |^|_ m e. ( ZZ>= ` n ) dom ( F ` m ) = U_ n e. Z |^|_ m e. ( ZZ>= ` n ) dom ( F ` m )
6	4, 5	allbutfi 39616	. . 3 |- ( X e. U_ n e. Z |^|_ m e. ( ZZ>= ` n ) dom ( F ` m ) <-> E. n e. Z A. m e. ( ZZ>= ` n ) X e. dom ( F ` m ) )
7	3, 6	sylib 208	. 2 |- ( ph -> E. n e. Z A. m e. ( ZZ>= ` n ) X e. dom ( F ` m ) )
8		allbutfifvre.1	. . . . 5 |- F/ m ph
9		nfv 1843	. . . . 5 |- F/ m n e. Z
10	8, 9	nfan 1828	. . . 4 |- F/ m ( ph /\ n e. Z )
11		simpll 790	. . . . 5 |- ( ( ( ph /\ n e. Z ) /\ m e. ( ZZ>= ` n ) ) -> ph )
12	4	uztrn2 11705	. . . . . . . 8 |- ( ( n e. Z /\ j e. ( ZZ>= ` n ) ) -> j e. Z )
13	12	ssd 39252	. . . . . . 7 |- ( n e. Z -> ( ZZ>= ` n ) C_ Z )
14	13	sselda 3603	. . . . . 6 |- ( ( n e. Z /\ m e. ( ZZ>= ` n ) ) -> m e. Z )
15	14	adantll 750	. . . . 5 |- ( ( ( ph /\ n e. Z ) /\ m e. ( ZZ>= ` n ) ) -> m e. Z )
16		allbutfifvre.3	. . . . . . 7 |- ( ( ph /\ m e. Z ) -> ( F ` m ) : dom ( F ` m ) --> RR )
17	16	ffvelrnda 6359	. . . . . 6 |- ( ( ( ph /\ m e. Z ) /\ X e. dom ( F ` m ) ) -> ( ( F ` m ) ` X ) e. RR )
18	17	ex 450	. . . . 5 |- ( ( ph /\ m e. Z ) -> ( X e. dom ( F ` m ) -> ( ( F ` m ) ` X ) e. RR ) )
19	11, 15, 18	syl2anc 693	. . . 4 |- ( ( ( ph /\ n e. Z ) /\ m e. ( ZZ>= ` n ) ) -> ( X e. dom ( F ` m ) -> ( ( F ` m ) ` X ) e. RR ) )
20	10, 19	ralimdaa 2958	. . 3 |- ( ( ph /\ n e. Z ) -> ( A. m e. ( ZZ>= ` n ) X e. dom ( F ` m ) -> A. m e. ( ZZ>= ` n ) ( ( F ` m ) ` X ) e. RR ) )
21	20	reximdva 3017	. 2 |- ( ph -> ( E. n e. Z A. m e. ( ZZ>= ` n ) X e. dom ( F ` m ) -> E. n e. Z A. m e. ( ZZ>= ` n ) ( ( F ` m ) ` X ) e. RR ) )
22	7, 21	mpd 15	1 |- ( ph -> E. n e. Z A. m e. ( ZZ>= ` n ) ( ( F ` m ) ` X ) e. RR )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   A.wral 2912   E.wrex 2913   U_ciun 4520   |^|_ciin 4521   dom cdm 5114   -->wf 5884   ` cfv 5888   RRcr 9935   ZZ>=cuz 11687

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-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  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-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  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-neg 10269  df-z 11378  df-uz 11688

This theorem is referenced by:  fnlimabslt  39911