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 ∈ (0..^(#‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))

Assertion

Ref	Expression
fargshiftfo	⊢ ((𝑁 ∈ ℕ0 ∧ 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 ∈ (0..^(#‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))
3	2	fargshiftf 41376	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → 𝐺:(0..^(#‘𝐹))⟶dom 𝐸)
4	1, 3	sylan2 491	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)–onto→dom 𝐸) → 𝐺:(0..^(#‘𝐹))⟶dom 𝐸)
5	2	rnmpt 5371	. . 3 ⊢ ran 𝐺 = {𝑦 ∣ ∃𝑥 ∈ (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 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)})
8	6, 7	syl 17	. . . . 5 ⊢ (𝐹:(1...𝑁)–onto→dom 𝐸 → ran 𝐹 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)})
9	8	adantl 482	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)–onto→dom 𝐸) → ran 𝐹 = {𝑦 ∣ ∃𝑧 ∈ (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 ∈ ℕ0 ∧ 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 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)} ↔ dom 𝐸 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)}))
14		eqcom 2629	. . . . . . . . 9 ⊢ (dom 𝐸 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)} ↔ {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)} = dom 𝐸)
15	13, 14	syl6bb 276	. . . . . . . 8 ⊢ (ran 𝐹 = dom 𝐸 → (ran 𝐹 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)} ↔ {𝑦 ∣ ∃𝑧 ∈ (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 ∈ ℕ0 ∧ 𝐹 Fn (1...𝑁)) → (#‘𝐹) = 𝑁)
18	16, 17	sylan2 491	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → (#‘𝐹) = 𝑁)
19	1, 18	sylan2 491	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)–onto→dom 𝐸) → (#‘𝐹) = 𝑁)
20		fz0add1fz1 12537	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ (0..^𝑁)) → (𝑥 + 1) ∈ (1...𝑁))
21		nn0z 11400	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ)
22		fzval3 12536	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℤ → (1...𝑁) = (1..^(𝑁 + 1)))
23	21, 22	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ0 → (1...𝑁) = (1..^(𝑁 + 1)))
24		nn0cn 11302	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ)
25		1cnd 10056	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ0 → 1 ∈ ℂ)
26	24, 25	addcomd 10238	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) = (1 + 𝑁))
27	26	oveq2d 6666	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ0 → (1..^(𝑁 + 1)) = (1..^(1 + 𝑁)))
28	23, 27	eqtrd 2656	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ0 → (1...𝑁) = (1..^(1 + 𝑁)))
29	28	eleq2d 2687	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ0 → (𝑧 ∈ (1...𝑁) ↔ 𝑧 ∈ (1..^(1 + 𝑁))))
30	29	biimpa 501	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑧 ∈ (1...𝑁)) → 𝑧 ∈ (1..^(1 + 𝑁)))
31	21	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑧 ∈ (1...𝑁)) → 𝑁 ∈ ℤ)
32		fzosubel3 12528	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ (1..^(1 + 𝑁)) ∧ 𝑁 ∈ ℤ) → (𝑧 − 1) ∈ (0..^𝑁))
33	30, 31, 32	syl2anc 693	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑧 ∈ (1...𝑁)) → (𝑧 − 1) ∈ (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 ∈ ℕ0 ∧ 𝑧 ∈ (1...𝑁)) ∧ 𝑥 = (𝑧 − 1)) → (𝑧 = (𝑥 + 1) ↔ 𝑧 = ((𝑧 − 1) + 1)))
37		elfzelz 12342	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ (1...𝑁) → 𝑧 ∈ ℤ)
38	37	zcnd 11483	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ (1...𝑁) → 𝑧 ∈ ℂ)
39	38	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑧 ∈ (1...𝑁)) → 𝑧 ∈ ℂ)
40		1cnd 10056	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑧 ∈ (1...𝑁)) → 1 ∈ ℂ)
41	39, 40	npcand 10396	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑧 ∈ (1...𝑁)) → ((𝑧 − 1) + 1) = 𝑧)
42	41	eqcomd 2628	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑧 ∈ (1...𝑁)) → 𝑧 = ((𝑧 − 1) + 1))
43	33, 36, 42	rspcedvd 3317	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑧 ∈ (1...𝑁)) → ∃𝑥 ∈ (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 ∈ ℕ0 ∧ 𝑧 = (𝑥 + 1)) → (𝑦 = (𝐹‘𝑧) ↔ 𝑦 = (𝐹‘(𝑥 + 1))))
47	20, 43, 46	rexxfrd 4881	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ0 → (∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧) ↔ ∃𝑥 ∈ (0..^𝑁)𝑦 = (𝐹‘(𝑥 + 1))))
48	47	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ (#‘𝐹) = 𝑁) → (∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧) ↔ ∃𝑥 ∈ (0..^𝑁)𝑦 = (𝐹‘(𝑥 + 1))))
49		oveq2 6658	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝐹) = 𝑁 → (0..^(#‘𝐹)) = (0..^𝑁))
50	49	rexeqdv 3145	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝐹) = 𝑁 → (∃𝑥 ∈ (0..^(#‘𝐹))𝑦 = (𝐹‘(𝑥 + 1)) ↔ ∃𝑥 ∈ (0..^𝑁)𝑦 = (𝐹‘(𝑥 + 1))))
51	50	bibi2d 332	. . . . . . . . . . . . . 14 ⊢ ((#‘𝐹) = 𝑁 → ((∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧) ↔ ∃𝑥 ∈ (0..^(#‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))) ↔ (∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧) ↔ ∃𝑥 ∈ (0..^𝑁)𝑦 = (𝐹‘(𝑥 + 1)))))
52	51	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ (#‘𝐹) = 𝑁) → ((∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧) ↔ ∃𝑥 ∈ (0..^(#‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))) ↔ (∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧) ↔ ∃𝑥 ∈ (0..^𝑁)𝑦 = (𝐹‘(𝑥 + 1)))))
53	48, 52	mpbird 247	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ (#‘𝐹) = 𝑁) → (∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧) ↔ ∃𝑥 ∈ (0..^(#‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))))
54	19, 53	syldan 487	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)–onto→dom 𝐸) → (∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧) ↔ ∃𝑥 ∈ (0..^(#‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))))
55	54	abbidv 2741	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)–onto→dom 𝐸) → {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)} = {𝑦 ∣ ∃𝑥 ∈ (0..^(#‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))})
56	55	eqeq1d 2624	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)–onto→dom 𝐸) → ({𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)} = dom 𝐸 ↔ {𝑦 ∣ ∃𝑥 ∈ (0..^(#‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))} = dom 𝐸))
57	56	biimpcd 239	. . . . . . . 8 ⊢ ({𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)} = dom 𝐸 → ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)–onto→dom 𝐸) → {𝑦 ∣ ∃𝑥 ∈ (0..^(#‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))} = dom 𝐸))
58	15, 57	syl6bi 243	. . . . . . 7 ⊢ (ran 𝐹 = dom 𝐸 → (ran 𝐹 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)} → ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)–onto→dom 𝐸) → {𝑦 ∣ ∃𝑥 ∈ (0..^(#‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))} = dom 𝐸)))
59	58	com23 86	. . . . . 6 ⊢ (ran 𝐹 = dom 𝐸 → ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)–onto→dom 𝐸) → (ran 𝐹 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)} → {𝑦 ∣ ∃𝑥 ∈ (0..^(#‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))} = dom 𝐸)))
60	59	adantl 482	. . . . 5 ⊢ ((𝐹 Fn (1...𝑁) ∧ ran 𝐹 = dom 𝐸) → ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)–onto→dom 𝐸) → (ran 𝐹 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)} → {𝑦 ∣ ∃𝑥 ∈ (0..^(#‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))} = dom 𝐸)))
61	12, 60	mpcom 38	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)–onto→dom 𝐸) → (ran 𝐹 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)} → {𝑦 ∣ ∃𝑥 ∈ (0..^(#‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))} = dom 𝐸))
62	9, 61	mpd 15	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)–onto→dom 𝐸) → {𝑦 ∣ ∃𝑥 ∈ (0..^(#‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))} = dom 𝐸)
63	5, 62	syl5eq 2668	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 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 ∈ ℕ0 ∧ 𝐹:(1...𝑁)–onto→dom 𝐸) → 𝐺:(0..^(#‘𝐹))–onto→dom 𝐸)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {cab 2608  ∃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  ℂcc 9934  0cc0 9936  1c1 9937   + caddc 9939   − cmin 10266  ℕ₀cn0 11292  ℤcz 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)