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Theorem fargshiftfo 41703
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 ∈ ℕ0𝐹:(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.
StepHypRef Expression
1 fof 6153 . . 3 (𝐹:(1...𝑁)–onto→dom 𝐸𝐹:(1...𝑁)⟶dom 𝐸)
2 fargshift.g . . . 4 𝐺 = (𝑥 ∈ (0..^(#‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))
32fargshiftf 41701 . . 3 ((𝑁 ∈ ℕ0𝐹:(1...𝑁)⟶dom 𝐸) → 𝐺:(0..^(#‘𝐹))⟶dom 𝐸)
41, 3sylan2 490 . 2 ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–onto→dom 𝐸) → 𝐺:(0..^(#‘𝐹))⟶dom 𝐸)
52rnmpt 5403 . . 3 ran 𝐺 = {𝑦 ∣ ∃𝑥 ∈ (0..^(#‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))}
6 fofn 6155 . . . . . 6 (𝐹:(1...𝑁)–onto→dom 𝐸𝐹 Fn (1...𝑁))
7 fnrnfv 6281 . . . . . 6 (𝐹 Fn (1...𝑁) → ran 𝐹 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧)})
86, 7syl 17 . . . . 5 (𝐹:(1...𝑁)–onto→dom 𝐸 → ran 𝐹 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧)})
98adantl 481 . . . 4 ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–onto→dom 𝐸) → ran 𝐹 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧)})
10 df-fo 5932 . . . . . . 7 (𝐹:(1...𝑁)–onto→dom 𝐸 ↔ (𝐹 Fn (1...𝑁) ∧ ran 𝐹 = dom 𝐸))
1110biimpi 206 . . . . . 6 (𝐹:(1...𝑁)–onto→dom 𝐸 → (𝐹 Fn (1...𝑁) ∧ ran 𝐹 = dom 𝐸))
1211adantl 481 . . . . 5 ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–onto→dom 𝐸) → (𝐹 Fn (1...𝑁) ∧ ran 𝐹 = dom 𝐸))
13 eqeq1 2655 . . . . . . . . 9 (ran 𝐹 = dom 𝐸 → (ran 𝐹 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧)} ↔ dom 𝐸 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧)}))
14 eqcom 2658 . . . . . . . . 9 (dom 𝐸 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧)} ↔ {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧)} = dom 𝐸)
1513, 14syl6bb 276 . . . . . . . 8 (ran 𝐹 = dom 𝐸 → (ran 𝐹 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧)} ↔ {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧)} = dom 𝐸))
16 ffn 6083 . . . . . . . . . . . . . 14 (𝐹:(1...𝑁)⟶dom 𝐸𝐹 Fn (1...𝑁))
17 fseq1hash 13203 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℕ0𝐹 Fn (1...𝑁)) → (#‘𝐹) = 𝑁)
1816, 17sylan2 490 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ0𝐹:(1...𝑁)⟶dom 𝐸) → (#‘𝐹) = 𝑁)
191, 18sylan2 490 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–onto→dom 𝐸) → (#‘𝐹) = 𝑁)
20 fz0add1fz1 12577 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℕ0𝑥 ∈ (0..^𝑁)) → (𝑥 + 1) ∈ (1...𝑁))
21 nn0z 11438 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0𝑁 ∈ ℤ)
22 fzval3 12576 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℤ → (1...𝑁) = (1..^(𝑁 + 1)))
2321, 22syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → (1...𝑁) = (1..^(𝑁 + 1)))
24 nn0cn 11340 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0𝑁 ∈ ℂ)
25 1cnd 10094 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0 → 1 ∈ ℂ)
2624, 25addcomd 10276 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → (𝑁 + 1) = (1 + 𝑁))
2726oveq2d 6706 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → (1..^(𝑁 + 1)) = (1..^(1 + 𝑁)))
2823, 27eqtrd 2685 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ0 → (1...𝑁) = (1..^(1 + 𝑁)))
2928eleq2d 2716 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ ℕ0 → (𝑧 ∈ (1...𝑁) ↔ 𝑧 ∈ (1..^(1 + 𝑁))))
3029biimpa 500 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ℕ0𝑧 ∈ (1...𝑁)) → 𝑧 ∈ (1..^(1 + 𝑁)))
3121adantr 480 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ℕ0𝑧 ∈ (1...𝑁)) → 𝑁 ∈ ℤ)
32 fzosubel3 12568 . . . . . . . . . . . . . . . . 17 ((𝑧 ∈ (1..^(1 + 𝑁)) ∧ 𝑁 ∈ ℤ) → (𝑧 − 1) ∈ (0..^𝑁))
3330, 31, 32syl2anc 694 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ0𝑧 ∈ (1...𝑁)) → (𝑧 − 1) ∈ (0..^𝑁))
34 oveq1 6697 . . . . . . . . . . . . . . . . . 18 (𝑥 = (𝑧 − 1) → (𝑥 + 1) = ((𝑧 − 1) + 1))
3534eqeq2d 2661 . . . . . . . . . . . . . . . . 17 (𝑥 = (𝑧 − 1) → (𝑧 = (𝑥 + 1) ↔ 𝑧 = ((𝑧 − 1) + 1)))
3635adantl 481 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0𝑧 ∈ (1...𝑁)) ∧ 𝑥 = (𝑧 − 1)) → (𝑧 = (𝑥 + 1) ↔ 𝑧 = ((𝑧 − 1) + 1)))
37 elfzelz 12380 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ (1...𝑁) → 𝑧 ∈ ℤ)
3837zcnd 11521 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ (1...𝑁) → 𝑧 ∈ ℂ)
3938adantl 481 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ ℕ0𝑧 ∈ (1...𝑁)) → 𝑧 ∈ ℂ)
40 1cnd 10094 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ ℕ0𝑧 ∈ (1...𝑁)) → 1 ∈ ℂ)
4139, 40npcand 10434 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ℕ0𝑧 ∈ (1...𝑁)) → ((𝑧 − 1) + 1) = 𝑧)
4241eqcomd 2657 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ0𝑧 ∈ (1...𝑁)) → 𝑧 = ((𝑧 − 1) + 1))
4333, 36, 42rspcedvd 3348 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℕ0𝑧 ∈ (1...𝑁)) → ∃𝑥 ∈ (0..^𝑁)𝑧 = (𝑥 + 1))
44 fveq2 6229 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑥 + 1) → (𝐹𝑧) = (𝐹‘(𝑥 + 1)))
4544eqeq2d 2661 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑥 + 1) → (𝑦 = (𝐹𝑧) ↔ 𝑦 = (𝐹‘(𝑥 + 1))))
4645adantl 481 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℕ0𝑧 = (𝑥 + 1)) → (𝑦 = (𝐹𝑧) ↔ 𝑦 = (𝐹‘(𝑥 + 1))))
4720, 43, 46rexxfrd 4911 . . . . . . . . . . . . . 14 (𝑁 ∈ ℕ0 → (∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧) ↔ ∃𝑥 ∈ (0..^𝑁)𝑦 = (𝐹‘(𝑥 + 1))))
4847adantr 480 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ0 ∧ (#‘𝐹) = 𝑁) → (∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧) ↔ ∃𝑥 ∈ (0..^𝑁)𝑦 = (𝐹‘(𝑥 + 1))))
49 oveq2 6698 . . . . . . . . . . . . . . . 16 ((#‘𝐹) = 𝑁 → (0..^(#‘𝐹)) = (0..^𝑁))
5049rexeqdv 3175 . . . . . . . . . . . . . . 15 ((#‘𝐹) = 𝑁 → (∃𝑥 ∈ (0..^(#‘𝐹))𝑦 = (𝐹‘(𝑥 + 1)) ↔ ∃𝑥 ∈ (0..^𝑁)𝑦 = (𝐹‘(𝑥 + 1))))
5150bibi2d 331 . . . . . . . . . . . . . 14 ((#‘𝐹) = 𝑁 → ((∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧) ↔ ∃𝑥 ∈ (0..^(#‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))) ↔ (∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧) ↔ ∃𝑥 ∈ (0..^𝑁)𝑦 = (𝐹‘(𝑥 + 1)))))
5251adantl 481 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ0 ∧ (#‘𝐹) = 𝑁) → ((∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧) ↔ ∃𝑥 ∈ (0..^(#‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))) ↔ (∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧) ↔ ∃𝑥 ∈ (0..^𝑁)𝑦 = (𝐹‘(𝑥 + 1)))))
5348, 52mpbird 247 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0 ∧ (#‘𝐹) = 𝑁) → (∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧) ↔ ∃𝑥 ∈ (0..^(#‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))))
5419, 53syldan 486 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–onto→dom 𝐸) → (∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧) ↔ ∃𝑥 ∈ (0..^(#‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))))
5554abbidv 2770 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–onto→dom 𝐸) → {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧)} = {𝑦 ∣ ∃𝑥 ∈ (0..^(#‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))})
5655eqeq1d 2653 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–onto→dom 𝐸) → ({𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧)} = dom 𝐸 ↔ {𝑦 ∣ ∃𝑥 ∈ (0..^(#‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))} = dom 𝐸))
5756biimpcd 239 . . . . . . . 8 ({𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧)} = dom 𝐸 → ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–onto→dom 𝐸) → {𝑦 ∣ ∃𝑥 ∈ (0..^(#‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))} = dom 𝐸))
5815, 57syl6bi 243 . . . . . . 7 (ran 𝐹 = dom 𝐸 → (ran 𝐹 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧)} → ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–onto→dom 𝐸) → {𝑦 ∣ ∃𝑥 ∈ (0..^(#‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))} = dom 𝐸)))
5958com23 86 . . . . . 6 (ran 𝐹 = dom 𝐸 → ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–onto→dom 𝐸) → (ran 𝐹 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧)} → {𝑦 ∣ ∃𝑥 ∈ (0..^(#‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))} = dom 𝐸)))
6059adantl 481 . . . . 5 ((𝐹 Fn (1...𝑁) ∧ ran 𝐹 = dom 𝐸) → ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–onto→dom 𝐸) → (ran 𝐹 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧)} → {𝑦 ∣ ∃𝑥 ∈ (0..^(#‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))} = dom 𝐸)))
6112, 60mpcom 38 . . . 4 ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–onto→dom 𝐸) → (ran 𝐹 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧)} → {𝑦 ∣ ∃𝑥 ∈ (0..^(#‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))} = dom 𝐸))
629, 61mpd 15 . . 3 ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–onto→dom 𝐸) → {𝑦 ∣ ∃𝑥 ∈ (0..^(#‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))} = dom 𝐸)
635, 62syl5eq 2697 . 2 ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–onto→dom 𝐸) → ran 𝐺 = dom 𝐸)
64 dffo2 6157 . 2 (𝐺:(0..^(#‘𝐹))–onto→dom 𝐸 ↔ (𝐺:(0..^(#‘𝐹))⟶dom 𝐸 ∧ ran 𝐺 = dom 𝐸))
654, 63, 64sylanbrc 699 1 ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–onto→dom 𝐸) → 𝐺:(0..^(#‘𝐹))–onto→dom 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  {cab 2637  wrex 2942  cmpt 4762  dom cdm 5143  ran crn 5144   Fn wfn 5921  wf 5922  ontowfo 5924  cfv 5926  (class class class)co 6690  cc 9972  0cc0 9974  1c1 9975   + caddc 9977  cmin 10304  0cn0 11330  cz 11415  ...cfz 12364  ..^cfzo 12504  #chash 13157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-hash 13158
This theorem is referenced by: (None)
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