Theorem iunincfi 39272

Description: Given a sequence of increasing sets, the union of a finite subsequence, is its last element. (Contributed by Glauco Siliprandi, 8-Apr-2021.)

Hypotheses

Ref	Expression
iunincfi.1	|- ( ph -> N e. ( ZZ>= ` M ) )
iunincfi.2	|- ( ( ph /\ n e. ( M..^ N ) ) -> ( F ` n ) C_ ( F ` ( n + 1 ) ) )

Assertion

Ref	Expression
iunincfi	|- ( ph -> U_ n e. ( M ... N ) ( F ` n ) = ( F ` N ) )

Distinct variable groups:   n, F   n, M   n, N   ph, n

Proof of Theorem iunincfi

Dummy variables m x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eliun 4524	. . . . . . 7 |- ( x e. U_ n e. ( M ... N ) ( F ` n ) <-> E. n e. ( M ... N ) x e. ( F ` n ) )
2	1	biimpi 206	. . . . . 6 |- ( x e. U_ n e. ( M ... N ) ( F ` n ) -> E. n e. ( M ... N ) x e. ( F ` n ) )
3	2	adantl 482	. . . . 5 |- ( ( ph /\ x e. U_ n e. ( M ... N ) ( F ` n ) ) -> E. n e. ( M ... N ) x e. ( F ` n ) )
4		elfzuz3 12339	. . . . . . . . . . . 12 |- ( n e. ( M ... N ) -> N e. ( ZZ>= ` n ) )
5	4	adantl 482	. . . . . . . . . . 11 |- ( ( ph /\ n e. ( M ... N ) ) -> N e. ( ZZ>= ` n ) )
6		simpll 790	. . . . . . . . . . . 12 |- ( ( ( ph /\ n e. ( M ... N ) ) /\ m e. ( n..^ N ) ) -> ph )
7		elfzuz 12338	. . . . . . . . . . . . . . . 16 |- ( n e. ( M ... N ) -> n e. ( ZZ>= ` M ) )
8		fzoss1 12495	. . . . . . . . . . . . . . . 16 |- ( n e. ( ZZ>= ` M ) -> ( n..^ N ) C_ ( M..^ N ) )
9	7, 8	syl 17	. . . . . . . . . . . . . . 15 |- ( n e. ( M ... N ) -> ( n..^ N ) C_ ( M..^ N ) )
10	9	adantr 481	. . . . . . . . . . . . . 14 |- ( ( n e. ( M ... N ) /\ m e. ( n..^ N ) ) -> ( n..^ N ) C_ ( M..^ N ) )
11		simpr 477	. . . . . . . . . . . . . 14 |- ( ( n e. ( M ... N ) /\ m e. ( n..^ N ) ) -> m e. ( n..^ N ) )
12	10, 11	sseldd 3604	. . . . . . . . . . . . 13 |- ( ( n e. ( M ... N ) /\ m e. ( n..^ N ) ) -> m e. ( M..^ N ) )
13	12	adantll 750	. . . . . . . . . . . 12 |- ( ( ( ph /\ n e. ( M ... N ) ) /\ m e. ( n..^ N ) ) -> m e. ( M..^ N ) )
14		eleq1 2689	. . . . . . . . . . . . . . 15 |- ( n = m -> ( n e. ( M..^ N ) <-> m e. ( M..^ N ) ) )
15	14	anbi2d 740	. . . . . . . . . . . . . 14 |- ( n = m -> ( ( ph /\ n e. ( M..^ N ) ) <-> ( ph /\ m e. ( M..^ N ) ) ) )
16		fveq2 6191	. . . . . . . . . . . . . . 15 |- ( n = m -> ( F ` n ) = ( F ` m ) )
17		oveq1 6657	. . . . . . . . . . . . . . . 16 |- ( n = m -> ( n + 1 ) = ( m + 1 ) )
18	17	fveq2d 6195	. . . . . . . . . . . . . . 15 |- ( n = m -> ( F ` ( n + 1 ) ) = ( F ` ( m + 1 ) ) )
19	16, 18	sseq12d 3634	. . . . . . . . . . . . . 14 |- ( n = m -> ( ( F ` n ) C_ ( F ` ( n + 1 ) ) <-> ( F ` m ) C_ ( F ` ( m + 1 ) ) ) )
20	15, 19	imbi12d 334	. . . . . . . . . . . . 13 |- ( n = m -> ( ( ( ph /\ n e. ( M..^ N ) ) -> ( F ` n ) C_ ( F ` ( n + 1 ) ) ) <-> ( ( ph /\ m e. ( M..^ N ) ) -> ( F ` m ) C_ ( F ` ( m + 1 ) ) ) ) )
21		iunincfi.2	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. ( M..^ N ) ) -> ( F ` n ) C_ ( F ` ( n + 1 ) ) )
22	20, 21	chvarv 2263	. . . . . . . . . . . 12 |- ( ( ph /\ m e. ( M..^ N ) ) -> ( F ` m ) C_ ( F ` ( m + 1 ) ) )
23	6, 13, 22	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( ph /\ n e. ( M ... N ) ) /\ m e. ( n..^ N ) ) -> ( F ` m ) C_ ( F ` ( m + 1 ) ) )
24	5, 23	ssinc 39264	. . . . . . . . . 10 |- ( ( ph /\ n e. ( M ... N ) ) -> ( F ` n ) C_ ( F ` N ) )
25	24	3adant3 1081	. . . . . . . . 9 |- ( ( ph /\ n e. ( M ... N ) /\ x e. ( F ` n ) ) -> ( F ` n ) C_ ( F ` N ) )
26		simp3 1063	. . . . . . . . 9 |- ( ( ph /\ n e. ( M ... N ) /\ x e. ( F ` n ) ) -> x e. ( F ` n ) )
27	25, 26	sseldd 3604	. . . . . . . 8 |- ( ( ph /\ n e. ( M ... N ) /\ x e. ( F ` n ) ) -> x e. ( F ` N ) )
28	27	3exp 1264	. . . . . . 7 |- ( ph -> ( n e. ( M ... N ) -> ( x e. ( F ` n ) -> x e. ( F ` N ) ) ) )
29	28	rexlimdv 3030	. . . . . 6 |- ( ph -> ( E. n e. ( M ... N ) x e. ( F ` n ) -> x e. ( F ` N ) ) )
30	29	imp 445	. . . . 5 |- ( ( ph /\ E. n e. ( M ... N ) x e. ( F ` n ) ) -> x e. ( F ` N ) )
31	3, 30	syldan 487	. . . 4 |- ( ( ph /\ x e. U_ n e. ( M ... N ) ( F ` n ) ) -> x e. ( F ` N ) )
32	31	ralrimiva 2966	. . 3 |- ( ph -> A. x e. U_ n e. ( M ... N ) ( F ` n ) x e. ( F ` N ) )
33		dfss3 3592	. . 3 |- ( U_ n e. ( M ... N ) ( F ` n ) C_ ( F ` N ) <-> A. x e. U_ n e. ( M ... N ) ( F ` n ) x e. ( F ` N ) )
34	32, 33	sylibr 224	. 2 |- ( ph -> U_ n e. ( M ... N ) ( F ` n ) C_ ( F ` N ) )
35		iunincfi.1	. . . 4 |- ( ph -> N e. ( ZZ>= ` M ) )
36		eluzfz2 12349	. . . 4 |- ( N e. ( ZZ>= ` M ) -> N e. ( M ... N ) )
37	35, 36	syl 17	. . 3 |- ( ph -> N e. ( M ... N ) )
38		fveq2 6191	. . . 4 |- ( n = N -> ( F ` n ) = ( F ` N ) )
39	38	ssiun2s 4564	. . 3 |- ( N e. ( M ... N ) -> ( F ` N ) C_ U_ n e. ( M ... N ) ( F ` n ) )
40	37, 39	syl 17	. 2 |- ( ph -> ( F ` N ) C_ U_ n e. ( M ... N ) ( F ` n ) )
41	34, 40	eqssd 3620	1 |- ( ph -> U_ n e. ( M ... N ) ( F ` n ) = ( F ` N ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   U_ciun 4520   ` cfv 5888  (class class class)co 6650   1c1 9937   + caddc 9939   ZZ>=cuz 11687   ...cfz 12326  ..^cfzo 12465

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

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-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  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-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466

This theorem is referenced by:  meaiuninclem  40697