Theorem fargshiftfo 41378

Description: If a function is onto, then also the shifted function is onto. (Contributed by Alexander van der Vekens, 24-Nov-2017.)

Hypothesis

Ref	Expression
fargshift.g	|- G = ( x e. ( 0..^ ( # ` F ) ) |-> ( F ` ( x + 1 ) ) )

Assertion

Ref	Expression
fargshiftfo	|- ( ( N e. NN0 /\ F : ( 1 ... N ) -onto-> dom E ) -> G : ( 0..^ ( # ` F ) ) -onto-> dom E )

Distinct variable groups:   x, F   x, E   x, N

Allowed substitution hint:   G( x)

Proof of Theorem fargshiftfo

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fof 6115	. . 3 |- ( F : ( 1 ... N ) -onto-> dom E -> F : ( 1 ... N ) --> dom E )
2		fargshift.g	. . . 4 |- G = ( x e. ( 0..^ ( # ` F ) ) |-> ( F ` ( x + 1 ) ) )
3	2	fargshiftf 41376	. . 3 |- ( ( N e. NN0 /\ F : ( 1 ... N ) --> dom E ) -> G : ( 0..^ ( # ` F ) ) --> dom E )
4	1, 3	sylan2 491	. 2 |- ( ( N e. NN0 /\ F : ( 1 ... N ) -onto-> dom E ) -> G : ( 0..^ ( # ` F ) ) --> dom E )
5	2	rnmpt 5371	. . 3 |- ran G = { y | E. x e. ( 0..^ ( # ` F ) ) y = ( F ` ( x + 1 ) ) }
6		fofn 6117	. . . . . 6 |- ( F : ( 1 ... N ) -onto-> dom E -> F Fn ( 1 ... N ) )
7		fnrnfv 6242	. . . . . 6 |- ( F Fn ( 1 ... N ) -> ran F = { y | E. z e. ( 1 ... N ) y = ( F ` z ) } )
8	6, 7	syl 17	. . . . 5 |- ( F : ( 1 ... N ) -onto-> dom E -> ran F = { y | E. z e. ( 1 ... N ) y = ( F ` z ) } )
9	8	adantl 482	. . . 4 |- ( ( N e. NN0 /\ F : ( 1 ... N ) -onto-> dom E ) -> ran F = { y | E. z e. ( 1 ... N ) y = ( F ` z ) } )
10		df-fo 5894	. . . . . . 7 |- ( F : ( 1 ... N ) -onto-> dom E <-> ( F Fn ( 1 ... N ) /\ ran F = dom E ) )
11	10	biimpi 206	. . . . . 6 |- ( F : ( 1 ... N ) -onto-> dom E -> ( F Fn ( 1 ... N ) /\ ran F = dom E ) )
12	11	adantl 482	. . . . 5 |- ( ( N e. NN0 /\ F : ( 1 ... N ) -onto-> dom E ) -> ( F Fn ( 1 ... N ) /\ ran F = dom E ) )
13		eqeq1 2626	. . . . . . . . 9 |- ( ran F = dom E -> ( ran F = { y | E. z e. ( 1 ... N ) y = ( F ` z ) } <-> dom E = { y | E. z e. ( 1 ... N ) y = ( F ` z ) } ) )
14		eqcom 2629	. . . . . . . . 9 |- ( dom E = { y | E. z e. ( 1 ... N ) y = ( F ` z ) } <-> { y | E. z e. ( 1 ... N ) y = ( F ` z ) } = dom E )
15	13, 14	syl6bb 276	. . . . . . . 8 |- ( ran F = dom E -> ( ran F = { y | E. z e. ( 1 ... N ) y = ( F ` z ) } <-> { y | E. z e. ( 1 ... N ) y = ( F ` z ) } = dom E ) )
16		ffn 6045	. . . . . . . . . . . . . 14 |- ( F : ( 1 ... N ) --> dom E -> F Fn ( 1 ... N ) )
17		fseq1hash 13165	. . . . . . . . . . . . . 14 |- ( ( N e. NN0 /\ F Fn ( 1 ... N ) ) -> ( # ` F ) = N )
18	16, 17	sylan2 491	. . . . . . . . . . . . 13 |- ( ( N e. NN0 /\ F : ( 1 ... N ) --> dom E ) -> ( # ` F ) = N )
19	1, 18	sylan2 491	. . . . . . . . . . . 12 |- ( ( N e. NN0 /\ F : ( 1 ... N ) -onto-> dom E ) -> ( # ` F ) = N )
20		fz0add1fz1 12537	. . . . . . . . . . . . . . 15 |- ( ( N e. NN0 /\ x e. ( 0..^ N ) ) -> ( x + 1 ) e. ( 1 ... N ) )
21		nn0z 11400	. . . . . . . . . . . . . . . . . . . . 21 |- ( N e. NN0 -> N e. ZZ )
22		fzval3 12536	. . . . . . . . . . . . . . . . . . . . 21 |- ( N e. ZZ -> ( 1 ... N ) = ( 1..^ ( N + 1 ) ) )
23	21, 22	syl 17	. . . . . . . . . . . . . . . . . . . 20 |- ( N e. NN0 -> ( 1 ... N ) = ( 1..^ ( N + 1 ) ) )
24		nn0cn 11302	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N e. NN0 -> N e. CC )
25		1cnd 10056	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N e. NN0 -> 1 e. CC )
26	24, 25	addcomd 10238	. . . . . . . . . . . . . . . . . . . . 21 |- ( N e. NN0 -> ( N + 1 ) = ( 1 + N ) )
27	26	oveq2d 6666	. . . . . . . . . . . . . . . . . . . 20 |- ( N e. NN0 -> ( 1..^ ( N + 1 ) ) = ( 1..^ ( 1 + N ) ) )
28	23, 27	eqtrd 2656	. . . . . . . . . . . . . . . . . . 19 |- ( N e. NN0 -> ( 1 ... N ) = ( 1..^ ( 1 + N ) ) )
29	28	eleq2d 2687	. . . . . . . . . . . . . . . . . 18 |- ( N e. NN0 -> ( z e. ( 1 ... N ) <-> z e. ( 1..^ ( 1 + N ) ) ) )
30	29	biimpa 501	. . . . . . . . . . . . . . . . 17 |- ( ( N e. NN0 /\ z e. ( 1 ... N ) ) -> z e. ( 1..^ ( 1 + N ) ) )
31	21	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( N e. NN0 /\ z e. ( 1 ... N ) ) -> N e. ZZ )
32		fzosubel3 12528	. . . . . . . . . . . . . . . . 17 |- ( ( z e. ( 1..^ ( 1 + N ) ) /\ N e. ZZ ) -> ( z - 1 ) e. ( 0..^ N ) )
33	30, 31, 32	syl2anc 693	. . . . . . . . . . . . . . . 16 |- ( ( N e. NN0 /\ z e. ( 1 ... N ) ) -> ( z - 1 ) e. ( 0..^ N ) )
34		oveq1 6657	. . . . . . . . . . . . . . . . . 18 |- ( x = ( z - 1 ) -> ( x + 1 ) = ( ( z - 1 ) + 1 ) )
35	34	eqeq2d 2632	. . . . . . . . . . . . . . . . 17 |- ( x = ( z - 1 ) -> ( z = ( x + 1 ) <-> z = ( ( z - 1 ) + 1 ) ) )
36	35	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( ( N e. NN0 /\ z e. ( 1 ... N ) ) /\ x = ( z - 1 ) ) -> ( z = ( x + 1 ) <-> z = ( ( z - 1 ) + 1 ) ) )
37		elfzelz 12342	. . . . . . . . . . . . . . . . . . . 20 |- ( z e. ( 1 ... N ) -> z e. ZZ )
38	37	zcnd 11483	. . . . . . . . . . . . . . . . . . 19 |- ( z e. ( 1 ... N ) -> z e. CC )
39	38	adantl 482	. . . . . . . . . . . . . . . . . 18 |- ( ( N e. NN0 /\ z e. ( 1 ... N ) ) -> z e. CC )
40		1cnd 10056	. . . . . . . . . . . . . . . . . 18 |- ( ( N e. NN0 /\ z e. ( 1 ... N ) ) -> 1 e. CC )
41	39, 40	npcand 10396	. . . . . . . . . . . . . . . . 17 |- ( ( N e. NN0 /\ z e. ( 1 ... N ) ) -> ( ( z - 1 ) + 1 ) = z )
42	41	eqcomd 2628	. . . . . . . . . . . . . . . 16 |- ( ( N e. NN0 /\ z e. ( 1 ... N ) ) -> z = ( ( z - 1 ) + 1 ) )
43	33, 36, 42	rspcedvd 3317	. . . . . . . . . . . . . . 15 |- ( ( N e. NN0 /\ z e. ( 1 ... N ) ) -> E. x e. ( 0..^ N ) z = ( x + 1 ) )
44		fveq2 6191	. . . . . . . . . . . . . . . . 17 |- ( z = ( x + 1 ) -> ( F ` z ) = ( F ` ( x + 1 ) ) )
45	44	eqeq2d 2632	. . . . . . . . . . . . . . . 16 |- ( z = ( x + 1 ) -> ( y = ( F ` z ) <-> y = ( F ` ( x + 1 ) ) ) )
46	45	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( N e. NN0 /\ z = ( x + 1 ) ) -> ( y = ( F ` z ) <-> y = ( F ` ( x + 1 ) ) ) )
47	20, 43, 46	rexxfrd 4881	. . . . . . . . . . . . . 14 |- ( N e. NN0 -> ( E. z e. ( 1 ... N ) y = ( F ` z ) <-> E. x e. ( 0..^ N ) y = ( F ` ( x + 1 ) ) ) )
48	47	adantr 481	. . . . . . . . . . . . 13 |- ( ( N e. NN0 /\ ( # ` F ) = N ) -> ( E. z e. ( 1 ... N ) y = ( F ` z ) <-> E. x e. ( 0..^ N ) y = ( F ` ( x + 1 ) ) ) )
49		oveq2 6658	. . . . . . . . . . . . . . . 16 |- ( ( # ` F ) = N -> ( 0..^ ( # ` F ) ) = ( 0..^ N ) )
50	49	rexeqdv 3145	. . . . . . . . . . . . . . 15 |- ( ( # ` F ) = N -> ( E. x e. ( 0..^ ( # ` F ) ) y = ( F ` ( x + 1 ) ) <-> E. x e. ( 0..^ N ) y = ( F ` ( x + 1 ) ) ) )
51	50	bibi2d 332	. . . . . . . . . . . . . 14 |- ( ( # ` F ) = N -> ( ( E. z e. ( 1 ... N ) y = ( F ` z ) <-> E. x e. ( 0..^ ( # ` F ) ) y = ( F ` ( x + 1 ) ) ) <-> ( E. z e. ( 1 ... N ) y = ( F ` z ) <-> E. x e. ( 0..^ N ) y = ( F ` ( x + 1 ) ) ) ) )
52	51	adantl 482	. . . . . . . . . . . . 13 |- ( ( N e. NN0 /\ ( # ` F ) = N ) -> ( ( E. z e. ( 1 ... N ) y = ( F ` z ) <-> E. x e. ( 0..^ ( # ` F ) ) y = ( F ` ( x + 1 ) ) ) <-> ( E. z e. ( 1 ... N ) y = ( F ` z ) <-> E. x e. ( 0..^ N ) y = ( F ` ( x + 1 ) ) ) ) )
53	48, 52	mpbird 247	. . . . . . . . . . . 12 |- ( ( N e. NN0 /\ ( # ` F ) = N ) -> ( E. z e. ( 1 ... N ) y = ( F ` z ) <-> E. x e. ( 0..^ ( # ` F ) ) y = ( F ` ( x + 1 ) ) ) )
54	19, 53	syldan 487	. . . . . . . . . . 11 |- ( ( N e. NN0 /\ F : ( 1 ... N ) -onto-> dom E ) -> ( E. z e. ( 1 ... N ) y = ( F ` z ) <-> E. x e. ( 0..^ ( # ` F ) ) y = ( F ` ( x + 1 ) ) ) )
55	54	abbidv 2741	. . . . . . . . . 10 |- ( ( N e. NN0 /\ F : ( 1 ... N ) -onto-> dom E ) -> { y | E. z e. ( 1 ... N ) y = ( F ` z ) } = { y | E. x e. ( 0..^ ( # ` F ) ) y = ( F ` ( x + 1 ) ) } )
56	55	eqeq1d 2624	. . . . . . . . 9 |- ( ( N e. NN0 /\ F : ( 1 ... N ) -onto-> dom E ) -> ( { y | E. z e. ( 1 ... N ) y = ( F ` z ) } = dom E <-> { y | E. x e. ( 0..^ ( # ` F ) ) y = ( F ` ( x + 1 ) ) } = dom E ) )
57	56	biimpcd 239	. . . . . . . 8 |- ( { y | E. z e. ( 1 ... N ) y = ( F ` z ) } = dom E -> ( ( N e. NN0 /\ F : ( 1 ... N ) -onto-> dom E ) -> { y | E. x e. ( 0..^ ( # ` F ) ) y = ( F ` ( x + 1 ) ) } = dom E ) )
58	15, 57	syl6bi 243	. . . . . . 7 |- ( ran F = dom E -> ( ran F = { y | E. z e. ( 1 ... N ) y = ( F ` z ) } -> ( ( N e. NN0 /\ F : ( 1 ... N ) -onto-> dom E ) -> { y | E. x e. ( 0..^ ( # ` F ) ) y = ( F ` ( x + 1 ) ) } = dom E ) ) )
59	58	com23 86	. . . . . 6 |- ( ran F = dom E -> ( ( N e. NN0 /\ F : ( 1 ... N ) -onto-> dom E ) -> ( ran F = { y | E. z e. ( 1 ... N ) y = ( F ` z ) } -> { y | E. x e. ( 0..^ ( # ` F ) ) y = ( F ` ( x + 1 ) ) } = dom E ) ) )
60	59	adantl 482	. . . . 5 |- ( ( F Fn ( 1 ... N ) /\ ran F = dom E ) -> ( ( N e. NN0 /\ F : ( 1 ... N ) -onto-> dom E ) -> ( ran F = { y | E. z e. ( 1 ... N ) y = ( F ` z ) } -> { y | E. x e. ( 0..^ ( # ` F ) ) y = ( F ` ( x + 1 ) ) } = dom E ) ) )
61	12, 60	mpcom 38	. . . 4 |- ( ( N e. NN0 /\ F : ( 1 ... N ) -onto-> dom E ) -> ( ran F = { y | E. z e. ( 1 ... N ) y = ( F ` z ) } -> { y | E. x e. ( 0..^ ( # ` F ) ) y = ( F ` ( x + 1 ) ) } = dom E ) )
62	9, 61	mpd 15	. . 3 |- ( ( N e. NN0 /\ F : ( 1 ... N ) -onto-> dom E ) -> { y | E. x e. ( 0..^ ( # ` F ) ) y = ( F ` ( x + 1 ) ) } = dom E )
63	5, 62	syl5eq 2668	. 2 |- ( ( N e. NN0 /\ F : ( 1 ... N ) -onto-> dom E ) -> ran G = dom E )
64		dffo2 6119	. 2 |- ( G : ( 0..^ ( # ` F ) ) -onto-> dom E <-> ( G : ( 0..^ ( # ` F ) ) --> dom E /\ ran G = dom E ) )
65	4, 63, 64	sylanbrc 698	1 |- ( ( N e. NN0 /\ F : ( 1 ... N ) -onto-> dom E ) -> G : ( 0..^ ( # ` F ) ) -onto-> dom E )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   |-> cmpt 4729   dom cdm 5114   ran crn 5115   Fn wfn 5883   -->wf 5884   -onto->wfo 5886   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   - cmin 10266   NN0cn0 11292   ZZcz 11377   ...cfz 12326  ..^cfzo 12465   #chash 13117

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-rep 4771  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-int 4476  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-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  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  df-hash 13118

This theorem is referenced by: (None)