Theorem fargshiftfo 47367

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 6821	. . 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 47365	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → 𝐺:(0..^(♯‘𝐹))⟶dom 𝐸)
4	1, 3	sylan2 593	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)–onto→dom 𝐸) → 𝐺:(0..^(♯‘𝐹))⟶dom 𝐸)
5	2	rnmpt 5971	. . 3 ⊢ ran 𝐺 = {𝑦 ∣ ∃𝑥 ∈ (0..^(♯‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))}
6		fofn 6823	. . . . . 6 ⊢ (𝐹:(1...𝑁)–onto→dom 𝐸 → 𝐹 Fn (1...𝑁))
7		fnrnfv 6968	. . . . . 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 481	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)–onto→dom 𝐸) → ran 𝐹 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)})
10		df-fo 6569	. . . . . . 7 ⊢ (𝐹:(1...𝑁)–onto→dom 𝐸 ↔ (𝐹 Fn (1...𝑁) ∧ ran 𝐹 = dom 𝐸))
11	10	biimpi 216	. . . . . 6 ⊢ (𝐹:(1...𝑁)–onto→dom 𝐸 → (𝐹 Fn (1...𝑁) ∧ ran 𝐹 = dom 𝐸))
12	11	adantl 481	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)–onto→dom 𝐸) → (𝐹 Fn (1...𝑁) ∧ ran 𝐹 = dom 𝐸))
13		eqeq1 2739	. . . . . . . . 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 2742	. . . . . . . . 9 ⊢ (dom 𝐸 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)} ↔ {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)} = dom 𝐸)
15	13, 14	bitrdi 287	. . . . . . . 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 6737	. . . . . . . . . . . . . 14 ⊢ (𝐹:(1...𝑁)⟶dom 𝐸 → 𝐹 Fn (1...𝑁))
17		fseq1hash 14412	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹 Fn (1...𝑁)) → (♯‘𝐹) = 𝑁)
18	16, 17	sylan2 593	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → (♯‘𝐹) = 𝑁)
19	1, 18	sylan2 593	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)–onto→dom 𝐸) → (♯‘𝐹) = 𝑁)
20		fz0add1fz1 13771	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ (0..^𝑁)) → (𝑥 + 1) ∈ (1...𝑁))
21		nn0z 12636	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ)
22		fzval3 13770	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℤ → (1...𝑁) = (1..^(𝑁 + 1)))
23	21, 22	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ0 → (1...𝑁) = (1..^(𝑁 + 1)))
24		nn0cn 12534	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ)
25		1cnd 11254	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ0 → 1 ∈ ℂ)
26	24, 25	addcomd 11461	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) = (1 + 𝑁))
27	26	oveq2d 7447	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ0 → (1..^(𝑁 + 1)) = (1..^(1 + 𝑁)))
28	23, 27	eqtrd 2775	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ0 → (1...𝑁) = (1..^(1 + 𝑁)))
29	28	eleq2d 2825	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ0 → (𝑧 ∈ (1...𝑁) ↔ 𝑧 ∈ (1..^(1 + 𝑁))))
30	29	biimpa 476	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑧 ∈ (1...𝑁)) → 𝑧 ∈ (1..^(1 + 𝑁)))
31	21	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑧 ∈ (1...𝑁)) → 𝑁 ∈ ℤ)
32		fzosubel3 13762	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ (1..^(1 + 𝑁)) ∧ 𝑁 ∈ ℤ) → (𝑧 − 1) ∈ (0..^𝑁))
33	30, 31, 32	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑧 ∈ (1...𝑁)) → (𝑧 − 1) ∈ (0..^𝑁))
34		oveq1 7438	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = (𝑧 − 1) → (𝑥 + 1) = ((𝑧 − 1) + 1))
35	34	eqeq2d 2746	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑧 − 1) → (𝑧 = (𝑥 + 1) ↔ 𝑧 = ((𝑧 − 1) + 1)))
36	35	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑧 ∈ (1...𝑁)) ∧ 𝑥 = (𝑧 − 1)) → (𝑧 = (𝑥 + 1) ↔ 𝑧 = ((𝑧 − 1) + 1)))
37		elfzelz 13561	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ (1...𝑁) → 𝑧 ∈ ℤ)
38	37	zcnd 12721	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ (1...𝑁) → 𝑧 ∈ ℂ)
39	38	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑧 ∈ (1...𝑁)) → 𝑧 ∈ ℂ)
40		1cnd 11254	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑧 ∈ (1...𝑁)) → 1 ∈ ℂ)
41	39, 40	npcand 11622	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑧 ∈ (1...𝑁)) → ((𝑧 − 1) + 1) = 𝑧)
42	41	eqcomd 2741	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑧 ∈ (1...𝑁)) → 𝑧 = ((𝑧 − 1) + 1))
43	33, 36, 42	rspcedvd 3624	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑧 ∈ (1...𝑁)) → ∃𝑥 ∈ (0..^𝑁)𝑧 = (𝑥 + 1))
44		fveq2 6907	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝑥 + 1) → (𝐹‘𝑧) = (𝐹‘(𝑥 + 1)))
45	44	eqeq2d 2746	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑥 + 1) → (𝑦 = (𝐹‘𝑧) ↔ 𝑦 = (𝐹‘(𝑥 + 1))))
46	45	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑧 = (𝑥 + 1)) → (𝑦 = (𝐹‘𝑧) ↔ 𝑦 = (𝐹‘(𝑥 + 1))))
47	20, 43, 46	rexxfrd 5415	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ0 → (∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧) ↔ ∃𝑥 ∈ (0..^𝑁)𝑦 = (𝐹‘(𝑥 + 1))))
48	47	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ (♯‘𝐹) = 𝑁) → (∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧) ↔ ∃𝑥 ∈ (0..^𝑁)𝑦 = (𝐹‘(𝑥 + 1))))
49		oveq2 7439	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝐹) = 𝑁 → (0..^(♯‘𝐹)) = (0..^𝑁))
50	49	rexeqdv 3325	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝐹) = 𝑁 → (∃𝑥 ∈ (0..^(♯‘𝐹))𝑦 = (𝐹‘(𝑥 + 1)) ↔ ∃𝑥 ∈ (0..^𝑁)𝑦 = (𝐹‘(𝑥 + 1))))
51	50	bibi2d 342	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . . 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 257	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ (♯‘𝐹) = 𝑁) → (∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧) ↔ ∃𝑥 ∈ (0..^(♯‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))))
54	19, 53	syldan 591	. . . . . . . . . . 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 2806	. . . . . . . . . 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 2737	. . . . . . . . 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 249	. . . . . . . 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	biimtrdi 253	. . . . . . 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 481	. . . . 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	eqtrid 2787	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)–onto→dom 𝐸) → ran 𝐺 = dom 𝐸)
64		dffo2 6825	. 2 ⊢ (𝐺:(0..^(♯‘𝐹))–onto→dom 𝐸 ↔ (𝐺:(0..^(♯‘𝐹))⟶dom 𝐸 ∧ ran 𝐺 = dom 𝐸))
65	4, 63, 64	sylanbrc 583	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)–onto→dom 𝐸) → 𝐺:(0..^(♯‘𝐹))–onto→dom 𝐸)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  {cab 2712  ∃wrex 3068   ↦ cmpt 5231  dom cdm 5689  ran crn 5690   Fn wfn 6558  ⟶wf 6559  –onto→wfo 6561  ‘cfv 6563  (class class class)co 7431  ℂcc 11151  0cc0 11153  1c1 11154   + caddc 11156   − cmin 11490  ℕ₀cn0 12524  ℤcz 12611  ...cfz 13544  ..^cfzo 13691  ♯chash 14366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-fzo 13692  df-hash 14367

This theorem is referenced by: (None)