Theorem iuninc 29379

Description: The union of an increasing collection of sets is its last element. (Contributed by Thierry Arnoux, 22-Jan-2017.)

Hypotheses

Ref	Expression
iuninc.1	|- ( ph -> F Fn NN )
iuninc.2	|- ( ( ph /\ n e. NN ) -> ( F ` n ) C_ ( F ` ( n + 1 ) ) )

Assertion

Ref	Expression
iuninc	|- ( ( ph /\ i e. NN ) -> U_ n e. ( 1 ... i ) ( F ` n ) = ( F ` i ) )

Distinct variable groups:   i, n   n, F   ph, n

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

Proof of Theorem iuninc

Dummy variables j k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 6658	. . . . . 6 |- ( j = 1 -> ( 1 ... j ) = ( 1 ... 1 ) )
2	1	iuneq1d 4545	. . . . 5 |- ( j = 1 -> U_ n e. ( 1 ... j ) ( F ` n ) = U_ n e. ( 1 ... 1 ) ( F ` n ) )
3		fveq2 6191	. . . . 5 |- ( j = 1 -> ( F ` j ) = ( F ` 1 ) )
4	2, 3	eqeq12d 2637	. . . 4 |- ( j = 1 -> ( U_ n e. ( 1 ... j ) ( F ` n ) = ( F ` j ) <-> U_ n e. ( 1 ... 1 ) ( F ` n ) = ( F ` 1 ) ) )
5	4	imbi2d 330	. . 3 |- ( j = 1 -> ( ( ph -> U_ n e. ( 1 ... j ) ( F ` n ) = ( F ` j ) ) <-> ( ph -> U_ n e. ( 1 ... 1 ) ( F ` n ) = ( F ` 1 ) ) ) )
6		oveq2 6658	. . . . . 6 |- ( j = k -> ( 1 ... j ) = ( 1 ... k ) )
7	6	iuneq1d 4545	. . . . 5 |- ( j = k -> U_ n e. ( 1 ... j ) ( F ` n ) = U_ n e. ( 1 ... k ) ( F ` n ) )
8		fveq2 6191	. . . . 5 |- ( j = k -> ( F ` j ) = ( F ` k ) )
9	7, 8	eqeq12d 2637	. . . 4 |- ( j = k -> ( U_ n e. ( 1 ... j ) ( F ` n ) = ( F ` j ) <-> U_ n e. ( 1 ... k ) ( F ` n ) = ( F ` k ) ) )
10	9	imbi2d 330	. . 3 |- ( j = k -> ( ( ph -> U_ n e. ( 1 ... j ) ( F ` n ) = ( F ` j ) ) <-> ( ph -> U_ n e. ( 1 ... k ) ( F ` n ) = ( F ` k ) ) ) )
11		oveq2 6658	. . . . . 6 |- ( j = ( k + 1 ) -> ( 1 ... j ) = ( 1 ... ( k + 1 ) ) )
12	11	iuneq1d 4545	. . . . 5 |- ( j = ( k + 1 ) -> U_ n e. ( 1 ... j ) ( F ` n ) = U_ n e. ( 1 ... ( k + 1 ) ) ( F ` n ) )
13		fveq2 6191	. . . . 5 |- ( j = ( k + 1 ) -> ( F ` j ) = ( F ` ( k + 1 ) ) )
14	12, 13	eqeq12d 2637	. . . 4 |- ( j = ( k + 1 ) -> ( U_ n e. ( 1 ... j ) ( F ` n ) = ( F ` j ) <-> U_ n e. ( 1 ... ( k + 1 ) ) ( F ` n ) = ( F ` ( k + 1 ) ) ) )
15	14	imbi2d 330	. . 3 |- ( j = ( k + 1 ) -> ( ( ph -> U_ n e. ( 1 ... j ) ( F ` n ) = ( F ` j ) ) <-> ( ph -> U_ n e. ( 1 ... ( k + 1 ) ) ( F ` n ) = ( F ` ( k + 1 ) ) ) ) )
16		oveq2 6658	. . . . . 6 |- ( j = i -> ( 1 ... j ) = ( 1 ... i ) )
17	16	iuneq1d 4545	. . . . 5 |- ( j = i -> U_ n e. ( 1 ... j ) ( F ` n ) = U_ n e. ( 1 ... i ) ( F ` n ) )
18		fveq2 6191	. . . . 5 |- ( j = i -> ( F ` j ) = ( F ` i ) )
19	17, 18	eqeq12d 2637	. . . 4 |- ( j = i -> ( U_ n e. ( 1 ... j ) ( F ` n ) = ( F ` j ) <-> U_ n e. ( 1 ... i ) ( F ` n ) = ( F ` i ) ) )
20	19	imbi2d 330	. . 3 |- ( j = i -> ( ( ph -> U_ n e. ( 1 ... j ) ( F ` n ) = ( F ` j ) ) <-> ( ph -> U_ n e. ( 1 ... i ) ( F ` n ) = ( F ` i ) ) ) )
21		1z 11407	. . . . . . 7 |- 1 e. ZZ
22		fzsn 12383	. . . . . . 7 |- ( 1 e. ZZ -> ( 1 ... 1 ) = { 1 } )
23	21, 22	ax-mp 5	. . . . . 6 |- ( 1 ... 1 ) = { 1 }
24		iuneq1 4534	. . . . . 6 |- ( ( 1 ... 1 ) = { 1 } -> U_ n e. ( 1 ... 1 ) ( F ` n ) = U_ n e. { 1 } ( F ` n ) )
25	23, 24	ax-mp 5	. . . . 5 |- U_ n e. ( 1 ... 1 ) ( F ` n ) = U_ n e. { 1 } ( F ` n )
26		1ex 10035	. . . . . 6 |- 1 e. _V
27		fveq2 6191	. . . . . 6 |- ( n = 1 -> ( F ` n ) = ( F ` 1 ) )
28	26, 27	iunxsn 4603	. . . . 5 |- U_ n e. { 1 } ( F ` n ) = ( F ` 1 )
29	25, 28	eqtri 2644	. . . 4 |- U_ n e. ( 1 ... 1 ) ( F ` n ) = ( F ` 1 )
30	29	a1i 11	. . 3 |- ( ph -> U_ n e. ( 1 ... 1 ) ( F ` n ) = ( F ` 1 ) )
31		simpll 790	. . . . . . 7 |- ( ( ( k e. NN /\ ph ) /\ U_ n e. ( 1 ... k ) ( F ` n ) = ( F ` k ) ) -> k e. NN )
32		elnnuz 11724	. . . . . . . . . 10 |- ( k e. NN <-> k e. ( ZZ>= ` 1 ) )
33		fzsuc 12388	. . . . . . . . . 10 |- ( k e. ( ZZ>= ` 1 ) -> ( 1 ... ( k + 1 ) ) = ( ( 1 ... k ) u. { ( k + 1 ) } ) )
34	32, 33	sylbi 207	. . . . . . . . 9 |- ( k e. NN -> ( 1 ... ( k + 1 ) ) = ( ( 1 ... k ) u. { ( k + 1 ) } ) )
35	34	iuneq1d 4545	. . . . . . . 8 |- ( k e. NN -> U_ n e. ( 1 ... ( k + 1 ) ) ( F ` n ) = U_ n e. ( ( 1 ... k ) u. { ( k + 1 ) } ) ( F ` n ) )
36		iunxun 4605	. . . . . . . . 9 |- U_ n e. ( ( 1 ... k ) u. { ( k + 1 ) } ) ( F ` n ) = ( U_ n e. ( 1 ... k ) ( F ` n ) u. U_ n e. { ( k + 1 ) } ( F ` n ) )
37		ovex 6678	. . . . . . . . . . 11 |- ( k + 1 ) e. _V
38		fveq2 6191	. . . . . . . . . . 11 |- ( n = ( k + 1 ) -> ( F ` n ) = ( F ` ( k + 1 ) ) )
39	37, 38	iunxsn 4603	. . . . . . . . . 10 |- U_ n e. { ( k + 1 ) } ( F ` n ) = ( F ` ( k + 1 ) )
40	39	uneq2i 3764	. . . . . . . . 9 |- ( U_ n e. ( 1 ... k ) ( F ` n ) u. U_ n e. { ( k + 1 ) } ( F ` n ) ) = ( U_ n e. ( 1 ... k ) ( F ` n ) u. ( F ` ( k + 1 ) ) )
41	36, 40	eqtri 2644	. . . . . . . 8 |- U_ n e. ( ( 1 ... k ) u. { ( k + 1 ) } ) ( F ` n ) = ( U_ n e. ( 1 ... k ) ( F ` n ) u. ( F ` ( k + 1 ) ) )
42	35, 41	syl6eq 2672	. . . . . . 7 |- ( k e. NN -> U_ n e. ( 1 ... ( k + 1 ) ) ( F ` n ) = ( U_ n e. ( 1 ... k ) ( F ` n ) u. ( F ` ( k + 1 ) ) ) )
43	31, 42	syl 17	. . . . . 6 |- ( ( ( k e. NN /\ ph ) /\ U_ n e. ( 1 ... k ) ( F ` n ) = ( F ` k ) ) -> U_ n e. ( 1 ... ( k + 1 ) ) ( F ` n ) = ( U_ n e. ( 1 ... k ) ( F ` n ) u. ( F ` ( k + 1 ) ) ) )
44		simpr 477	. . . . . . 7 |- ( ( ( k e. NN /\ ph ) /\ U_ n e. ( 1 ... k ) ( F ` n ) = ( F ` k ) ) -> U_ n e. ( 1 ... k ) ( F ` n ) = ( F ` k ) )
45	44	uneq1d 3766	. . . . . 6 |- ( ( ( k e. NN /\ ph ) /\ U_ n e. ( 1 ... k ) ( F ` n ) = ( F ` k ) ) -> ( U_ n e. ( 1 ... k ) ( F ` n ) u. ( F ` ( k + 1 ) ) ) = ( ( F ` k ) u. ( F ` ( k + 1 ) ) ) )
46		simplr 792	. . . . . . 7 |- ( ( ( k e. NN /\ ph ) /\ U_ n e. ( 1 ... k ) ( F ` n ) = ( F ` k ) ) -> ph )
47		iuninc.2	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> ( F ` n ) C_ ( F ` ( n + 1 ) ) )
48	47	sbt 2419	. . . . . . . . 9 |- [ k / n ] ( ( ph /\ n e. NN ) -> ( F ` n ) C_ ( F ` ( n + 1 ) ) )
49		sbim 2395	. . . . . . . . . 10 |- ( [ k / n ] ( ( ph /\ n e. NN ) -> ( F ` n ) C_ ( F ` ( n + 1 ) ) ) <-> ( [ k / n ] ( ph /\ n e. NN ) -> [ k / n ] ( F ` n ) C_ ( F ` ( n + 1 ) ) ) )
50		sban 2399	. . . . . . . . . . . 12 |- ( [ k / n ] ( ph /\ n e. NN ) <-> ( [ k / n ] ph /\ [ k / n ] n e. NN ) )
51		nfv 1843	. . . . . . . . . . . . . 14 |- F/ n ph
52	51	sbf 2380	. . . . . . . . . . . . 13 |- ( [ k / n ] ph <-> ph )
53		clelsb3 2729	. . . . . . . . . . . . 13 |- ( [ k / n ] n e. NN <-> k e. NN )
54	52, 53	anbi12i 733	. . . . . . . . . . . 12 |- ( ( [ k / n ] ph /\ [ k / n ] n e. NN ) <-> ( ph /\ k e. NN ) )
55	50, 54	bitr2i 265	. . . . . . . . . . 11 |- ( ( ph /\ k e. NN ) <-> [ k / n ] ( ph /\ n e. NN ) )
56		sbsbc 3439	. . . . . . . . . . . 12 |- ( [ k / n ] ( F ` n ) C_ ( F ` ( n + 1 ) ) <-> [. k / n ]. ( F ` n ) C_ ( F ` ( n + 1 ) ) )
57		vex 3203	. . . . . . . . . . . . 13 |- k e. _V
58		sbcssg 4085	. . . . . . . . . . . . 13 |- ( k e. _V -> ( [. k / n ]. ( F ` n ) C_ ( F ` ( n + 1 ) ) <-> [_ k / n ]_ ( F ` n ) C_ [_ k / n ]_ ( F ` ( n + 1 ) ) ) )
59	57, 58	ax-mp 5	. . . . . . . . . . . 12 |- ( [. k / n ]. ( F ` n ) C_ ( F ` ( n + 1 ) ) <-> [_ k / n ]_ ( F ` n ) C_ [_ k / n ]_ ( F ` ( n + 1 ) ) )
60		csbfv 6233	. . . . . . . . . . . . 13 |- [_ k / n ]_ ( F ` n ) = ( F ` k )
61		csbfv2g 6232	. . . . . . . . . . . . . . 15 |- ( k e. _V -> [_ k / n ]_ ( F ` ( n + 1 ) ) = ( F ` [_ k / n ]_ ( n + 1 ) ) )
62	57, 61	ax-mp 5	. . . . . . . . . . . . . 14 |- [_ k / n ]_ ( F ` ( n + 1 ) ) = ( F ` [_ k / n ]_ ( n + 1 ) )
63		csbov1g 6690	. . . . . . . . . . . . . . . 16 |- ( k e. _V -> [_ k / n ]_ ( n + 1 ) = ( [_ k / n ]_ n + 1 ) )
64	57, 63	ax-mp 5	. . . . . . . . . . . . . . 15 |- [_ k / n ]_ ( n + 1 ) = ( [_ k / n ]_ n + 1 )
65	64	fveq2i 6194	. . . . . . . . . . . . . 14 |- ( F ` [_ k / n ]_ ( n + 1 ) ) = ( F ` ( [_ k / n ]_ n + 1 ) )
66		csbvarg 4003	. . . . . . . . . . . . . . . . 17 |- ( k e. _V -> [_ k / n ]_ n = k )
67	57, 66	ax-mp 5	. . . . . . . . . . . . . . . 16 |- [_ k / n ]_ n = k
68	67	oveq1i 6660	. . . . . . . . . . . . . . 15 |- ( [_ k / n ]_ n + 1 ) = ( k + 1 )
69	68	fveq2i 6194	. . . . . . . . . . . . . 14 |- ( F ` ( [_ k / n ]_ n + 1 ) ) = ( F ` ( k + 1 ) )
70	62, 65, 69	3eqtri 2648	. . . . . . . . . . . . 13 |- [_ k / n ]_ ( F ` ( n + 1 ) ) = ( F ` ( k + 1 ) )
71	60, 70	sseq12i 3631	. . . . . . . . . . . 12 |- ( [_ k / n ]_ ( F ` n ) C_ [_ k / n ]_ ( F ` ( n + 1 ) ) <-> ( F ` k ) C_ ( F ` ( k + 1 ) ) )
72	56, 59, 71	3bitrri 287	. . . . . . . . . . 11 |- ( ( F ` k ) C_ ( F ` ( k + 1 ) ) <-> [ k / n ] ( F ` n ) C_ ( F ` ( n + 1 ) ) )
73	55, 72	imbi12i 340	. . . . . . . . . 10 |- ( ( ( ph /\ k e. NN ) -> ( F ` k ) C_ ( F ` ( k + 1 ) ) ) <-> ( [ k / n ] ( ph /\ n e. NN ) -> [ k / n ] ( F ` n ) C_ ( F ` ( n + 1 ) ) ) )
74	49, 73	bitr4i 267	. . . . . . . . 9 |- ( [ k / n ] ( ( ph /\ n e. NN ) -> ( F ` n ) C_ ( F ` ( n + 1 ) ) ) <-> ( ( ph /\ k e. NN ) -> ( F ` k ) C_ ( F ` ( k + 1 ) ) ) )
75	48, 74	mpbi 220	. . . . . . . 8 |- ( ( ph /\ k e. NN ) -> ( F ` k ) C_ ( F ` ( k + 1 ) ) )
76		ssequn1 3783	. . . . . . . 8 |- ( ( F ` k ) C_ ( F ` ( k + 1 ) ) <-> ( ( F ` k ) u. ( F ` ( k + 1 ) ) ) = ( F ` ( k + 1 ) ) )
77	75, 76	sylib 208	. . . . . . 7 |- ( ( ph /\ k e. NN ) -> ( ( F ` k ) u. ( F ` ( k + 1 ) ) ) = ( F ` ( k + 1 ) ) )
78	46, 31, 77	syl2anc 693	. . . . . 6 |- ( ( ( k e. NN /\ ph ) /\ U_ n e. ( 1 ... k ) ( F ` n ) = ( F ` k ) ) -> ( ( F ` k ) u. ( F ` ( k + 1 ) ) ) = ( F ` ( k + 1 ) ) )
79	43, 45, 78	3eqtrd 2660	. . . . 5 |- ( ( ( k e. NN /\ ph ) /\ U_ n e. ( 1 ... k ) ( F ` n ) = ( F ` k ) ) -> U_ n e. ( 1 ... ( k + 1 ) ) ( F ` n ) = ( F ` ( k + 1 ) ) )
80	79	exp31 630	. . . 4 |- ( k e. NN -> ( ph -> ( U_ n e. ( 1 ... k ) ( F ` n ) = ( F ` k ) -> U_ n e. ( 1 ... ( k + 1 ) ) ( F ` n ) = ( F ` ( k + 1 ) ) ) ) )
81	80	a2d 29	. . 3 |- ( k e. NN -> ( ( ph -> U_ n e. ( 1 ... k ) ( F ` n ) = ( F ` k ) ) -> ( ph -> U_ n e. ( 1 ... ( k + 1 ) ) ( F ` n ) = ( F ` ( k + 1 ) ) ) ) )
82	5, 10, 15, 20, 30, 81	nnind 11038	. 2 |- ( i e. NN -> ( ph -> U_ n e. ( 1 ... i ) ( F ` n ) = ( F ` i ) ) )
83	82	impcom 446	1 |- ( ( ph /\ i e. NN ) -> U_ n e. ( 1 ... i ) ( F ` n ) = ( F ` i ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   [wsb 1880   e. wcel 1990   _Vcvv 3200   [.wsbc 3435   [_csb 3533   u. cun 3572   C_ wss 3574   {csn 4177   U_ciun 4520   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   1c1 9937   + caddc 9939   NNcn 11020   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326

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-fal 1489  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

This theorem is referenced by:  meascnbl  30282