Theorem fargshiftfo 48729

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)⟶ontodom 𝐸) → 𝐺:(0..^(♯‘𝐹))⟶ontodom 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 6818	. . 3 ⊢ (𝐹:(1...𝑁)⟶ontodom 𝐸 → 𝐹:(1...𝑁)⟶dom 𝐸)
2		fargshift.g	. . . 4 ⊢ 𝐺 = (𝑥 ∈ (0..^(♯‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))
3	2	fargshiftf 48727	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → 𝐺:(0..^(♯‘𝐹))⟶dom 𝐸)
4	1, 3	sylan2 605	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶ontodom 𝐸) → 𝐺:(0..^(♯‘𝐹))⟶dom 𝐸)
5	2	rnmpt 5946	. . 3 ⊢ ran 𝐺 = {𝑦 ∣ ∃𝑥 ∈ (0..^(♯‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))}
6		fofn 6820	. . . . . 6 ⊢ (𝐹:(1...𝑁)⟶ontodom 𝐸 → 𝐹 Fn (1...𝑁))
7		fnrnfv 6967	. . . . . 6 ⊢ (𝐹 Fn (1...𝑁) → ran 𝐹 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)})
8	6, 7	syl 18	. . . . 5 ⊢ (𝐹:(1...𝑁)⟶ontodom 𝐸 → ran 𝐹 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)})
9	8	adantl 487	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶ontodom 𝐸) → ran 𝐹 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)})
10		df-fo 6564	. . . . . 6 ⊢ (𝐹:(1...𝑁)⟶ontodom 𝐸 ↔ (𝐹 Fn (1...𝑁) ∧ ran 𝐹 = dom 𝐸))
11	10	bilani 510	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶ontodom 𝐸) → (𝐹 Fn (1...𝑁) ∧ ran 𝐹 = dom 𝐸))
12		eqeq1 2767	. . . . . . . . 9 ⊢ (ran 𝐹 = dom 𝐸 → (ran 𝐹 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)} ↔ dom 𝐸 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)}))
13		eqcom 2770	. . . . . . . . 9 ⊢ (dom 𝐸 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)} ↔ {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)} = dom 𝐸)
14	12, 13	bitrdi 290	. . . . . . . 8 ⊢ (ran 𝐹 = dom 𝐸 → (ran 𝐹 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)} ↔ {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)} = dom 𝐸))
15		ffn 6731	. . . . . . . . . . . . . 14 ⊢ (𝐹:(1...𝑁)⟶dom 𝐸 → 𝐹 Fn (1...𝑁))
16		fseq1hash 14547	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹 Fn (1...𝑁)) → (♯‘𝐹) = 𝑁)
17	15, 16	sylan2 605	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → (♯‘𝐹) = 𝑁)
18	1, 17	sylan2 605	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶ontodom 𝐸) → (♯‘𝐹) = 𝑁)
19		fz0add1fz1 13897	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ (0..^𝑁)) → (𝑥 + 1) ∈ (1...𝑁))
20		nn0z 12744	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ)
21		fzval3 13896	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℤ → (1...𝑁) = (1..^(𝑁 + 1)))
22	20, 21	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ0 → (1...𝑁) = (1..^(𝑁 + 1)))
23		nn0cn 12643	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ)
24		1cnd 11329	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ0 → 1 ∈ ℂ)
25	23, 24	addcomd 11539	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) = (1 + 𝑁))
26	25	oveq2d 7459	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ0 → (1..^(𝑁 + 1)) = (1..^(1 + 𝑁)))
27	22, 26	eqtrd 2798	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ0 → (1...𝑁) = (1..^(1 + 𝑁)))
28	27	eleq2d 2850	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ0 → (𝑧 ∈ (1...𝑁) ↔ 𝑧 ∈ (1..^(1 + 𝑁))))
29	28	biimpa 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑧 ∈ (1...𝑁)) → 𝑧 ∈ (1..^(1 + 𝑁)))
30	20	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑧 ∈ (1...𝑁)) → 𝑁 ∈ ℤ)
31		fzosubel3 13888	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ (1..^(1 + 𝑁)) ∧ 𝑁 ∈ ℤ) → (𝑧 − 1) ∈ (0..^𝑁))
32	29, 30, 31	syl2anc 596	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑧 ∈ (1...𝑁)) → (𝑧 − 1) ∈ (0..^𝑁))
33		oveq1 7450	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = (𝑧 − 1) → (𝑥 + 1) = ((𝑧 − 1) + 1))
34	33	eqeq2d 2774	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑧 − 1) → (𝑧 = (𝑥 + 1) ↔ 𝑧 = ((𝑧 − 1) + 1)))
35	34	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑧 ∈ (1...𝑁)) ∧ 𝑥 = (𝑧 − 1)) → (𝑧 = (𝑥 + 1) ↔ 𝑧 = ((𝑧 − 1) + 1)))
36		elfzelz 13683	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ (1...𝑁) → 𝑧 ∈ ℤ)
37	36	zcnd 12831	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ (1...𝑁) → 𝑧 ∈ ℂ)
38	37	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑧 ∈ (1...𝑁)) → 𝑧 ∈ ℂ)
39		1cnd 11329	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑧 ∈ (1...𝑁)) → 1 ∈ ℂ)
40	38, 39	npcand 11700	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑧 ∈ (1...𝑁)) → ((𝑧 − 1) + 1) = 𝑧)
41	40	eqcomd 2769	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑧 ∈ (1...𝑁)) → 𝑧 = ((𝑧 − 1) + 1))
42	32, 35, 41	rspcedvd 3582	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑧 ∈ (1...𝑁)) → ∃𝑥 ∈ (0..^𝑁)𝑧 = (𝑥 + 1))
43		fveq2 6908	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝑥 + 1) → (𝐹‘𝑧) = (𝐹‘(𝑥 + 1)))
44	43	eqeq2d 2774	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑥 + 1) → (𝑦 = (𝐹‘𝑧) ↔ 𝑦 = (𝐹‘(𝑥 + 1))))
45	44	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑧 = (𝑥 + 1)) → (𝑦 = (𝐹‘𝑧) ↔ 𝑦 = (𝐹‘(𝑥 + 1))))
46	19, 42, 45	rexxfrd 5375	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ0 → (∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧) ↔ ∃𝑥 ∈ (0..^𝑁)𝑦 = (𝐹‘(𝑥 + 1))))
47	46	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ (♯‘𝐹) = 𝑁) → (∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧) ↔ ∃𝑥 ∈ (0..^𝑁)𝑦 = (𝐹‘(𝑥 + 1))))
48		oveq2 7451	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝐹) = 𝑁 → (0..^(♯‘𝐹)) = (0..^𝑁))
49	48	rexeqdv 3324	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝐹) = 𝑁 → (∃𝑥 ∈ (0..^(♯‘𝐹))𝑦 = (𝐹‘(𝑥 + 1)) ↔ ∃𝑥 ∈ (0..^𝑁)𝑦 = (𝐹‘(𝑥 + 1))))
50	49	bibi2d 345	. . . . . . . . . . . . . 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)))))
51	50	adantl 487	. . . . . . . . . . . . 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)))))
52	47, 51	mpbird 260	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ (♯‘𝐹) = 𝑁) → (∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧) ↔ ∃𝑥 ∈ (0..^(♯‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))))
53	18, 52	syldan 603	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶ontodom 𝐸) → (∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧) ↔ ∃𝑥 ∈ (0..^(♯‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))))
54	53	abbidv 2829	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶ontodom 𝐸) → {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)} = {𝑦 ∣ ∃𝑥 ∈ (0..^(♯‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))})
55	54	eqeq1d 2765	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶ontodom 𝐸) → ({𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)} = dom 𝐸 ↔ {𝑦 ∣ ∃𝑥 ∈ (0..^(♯‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))} = dom 𝐸))
56	55	biimpcd 252	. . . . . . . 8 ⊢ ({𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)} = dom 𝐸 → ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶ontodom 𝐸) → {𝑦 ∣ ∃𝑥 ∈ (0..^(♯‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))} = dom 𝐸))
57	14, 56	biimtrdi 256	. . . . . . 7 ⊢ (ran 𝐹 = dom 𝐸 → (ran 𝐹 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)} → ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶ontodom 𝐸) → {𝑦 ∣ ∃𝑥 ∈ (0..^(♯‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))} = dom 𝐸)))
58	57	com23 87	. . . . . 6 ⊢ (ran 𝐹 = dom 𝐸 → ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶ontodom 𝐸) → (ran 𝐹 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)} → {𝑦 ∣ ∃𝑥 ∈ (0..^(♯‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))} = dom 𝐸)))
59	58	adantl 487	. . . . 5 ⊢ ((𝐹 Fn (1...𝑁) ∧ ran 𝐹 = dom 𝐸) → ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶ontodom 𝐸) → (ran 𝐹 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)} → {𝑦 ∣ ∃𝑥 ∈ (0..^(♯‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))} = dom 𝐸)))
60	11, 59	mpcom 39	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶ontodom 𝐸) → (ran 𝐹 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)} → {𝑦 ∣ ∃𝑥 ∈ (0..^(♯‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))} = dom 𝐸))
61	9, 60	mpd 16	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶ontodom 𝐸) → {𝑦 ∣ ∃𝑥 ∈ (0..^(♯‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))} = dom 𝐸)
62	5, 61	eqtrid 2810	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶ontodom 𝐸) → ran 𝐺 = dom 𝐸)
63		dffo2 6822	. 2 ⊢ (𝐺:(0..^(♯‘𝐹))⟶ontodom 𝐸 ↔ (𝐺:(0..^(♯‘𝐹))⟶dom 𝐸 ∧ ran 𝐺 = dom 𝐸))
64	4, 62, 63	sylanbrc 595	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶ontodom 𝐸) → 𝐺:(0..^(♯‘𝐹))⟶ontodom 𝐸)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {cab 2741  ∃wrex 3090   ↦ cmpt 5189  dom cdm 5655  ran crn 5656   Fn wfn 6553  ⟶wf 6554  ⟶ontowfo 6556  ‘cfv 6558  (class class class)co 7443  ℂcc 11225  0cc0 11227  1c1 11228   + caddc 11230   − cmin 11568  ℕ₀cn0 12633  ℤcz 12720  ...cfz 13666  ..^cfzo 13815  ♯chash 14501

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-cnex 11283  ax-resscn 11284  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-addrcl 11288  ax-mulcl 11289  ax-mulrcl 11290  ax-mulcom 11291  ax-addass 11292  ax-mulass 11293  ax-distr 11294  ax-i2m1 11295  ax-1ne0 11296  ax-1rid 11297  ax-rnegex 11298  ax-rrecex 11299  ax-cnre 11300  ax-pre-lttri 11301  ax-pre-lttrn 11302  ax-pre-ltadd 11303  ax-pre-mulgt0 11304

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-er 8745  df-en 9002  df-dom 9003  df-sdom 9004  df-fin 9005  df-card 10047  df-pnf 11372  df-mnf 11373  df-xr 11374  df-ltxr 11375  df-le 11376  df-sub 11570  df-neg 11571  df-nn 12363  df-n0 12634  df-z 12721  df-uz 12993  df-fz 13667  df-fzo 13816  df-hash 14502

This theorem is used by: (None)