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Theorem fargshiftfo 43038
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 6450 . . 3 (𝐹:(1...𝑁)–onto→dom 𝐸𝐹:(1...𝑁)⟶dom 𝐸)
2 fargshift.g . . . 4 𝐺 = (𝑥 ∈ (0..^(♯‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))
32fargshiftf 43036 . . 3 ((𝑁 ∈ ℕ0𝐹:(1...𝑁)⟶dom 𝐸) → 𝐺:(0..^(♯‘𝐹))⟶dom 𝐸)
41, 3sylan2 592 . 2 ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–onto→dom 𝐸) → 𝐺:(0..^(♯‘𝐹))⟶dom 𝐸)
52rnmpt 5701 . . 3 ran 𝐺 = {𝑦 ∣ ∃𝑥 ∈ (0..^(♯‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))}
6 fofn 6452 . . . . . 6 (𝐹:(1...𝑁)–onto→dom 𝐸𝐹 Fn (1...𝑁))
7 fnrnfv 6585 . . . . . 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 482 . . . 4 ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–onto→dom 𝐸) → ran 𝐹 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧)})
10 df-fo 6223 . . . . . . 7 (𝐹:(1...𝑁)–onto→dom 𝐸 ↔ (𝐹 Fn (1...𝑁) ∧ ran 𝐹 = dom 𝐸))
1110biimpi 217 . . . . . 6 (𝐹:(1...𝑁)–onto→dom 𝐸 → (𝐹 Fn (1...𝑁) ∧ ran 𝐹 = dom 𝐸))
1211adantl 482 . . . . 5 ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–onto→dom 𝐸) → (𝐹 Fn (1...𝑁) ∧ ran 𝐹 = dom 𝐸))
13 eqeq1 2797 . . . . . . . . 9 (ran 𝐹 = dom 𝐸 → (ran 𝐹 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧)} ↔ dom 𝐸 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧)}))
14 eqcom 2800 . . . . . . . . 9 (dom 𝐸 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧)} ↔ {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧)} = dom 𝐸)
1513, 14syl6bb 288 . . . . . . . 8 (ran 𝐹 = dom 𝐸 → (ran 𝐹 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧)} ↔ {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧)} = dom 𝐸))
16 ffn 6374 . . . . . . . . . . . . . 14 (𝐹:(1...𝑁)⟶dom 𝐸𝐹 Fn (1...𝑁))
17 fseq1hash 13573 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℕ0𝐹 Fn (1...𝑁)) → (♯‘𝐹) = 𝑁)
1816, 17sylan2 592 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ0𝐹:(1...𝑁)⟶dom 𝐸) → (♯‘𝐹) = 𝑁)
191, 18sylan2 592 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–onto→dom 𝐸) → (♯‘𝐹) = 𝑁)
20 fz0add1fz1 12945 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℕ0𝑥 ∈ (0..^𝑁)) → (𝑥 + 1) ∈ (1...𝑁))
21 nn0z 11843 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0𝑁 ∈ ℤ)
22 fzval3 12944 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℤ → (1...𝑁) = (1..^(𝑁 + 1)))
2321, 22syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → (1...𝑁) = (1..^(𝑁 + 1)))
24 nn0cn 11744 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0𝑁 ∈ ℂ)
25 1cnd 10471 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0 → 1 ∈ ℂ)
2624, 25addcomd 10678 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → (𝑁 + 1) = (1 + 𝑁))
2726oveq2d 7023 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → (1..^(𝑁 + 1)) = (1..^(1 + 𝑁)))
2823, 27eqtrd 2829 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ0 → (1...𝑁) = (1..^(1 + 𝑁)))
2928eleq2d 2866 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ ℕ0 → (𝑧 ∈ (1...𝑁) ↔ 𝑧 ∈ (1..^(1 + 𝑁))))
3029biimpa 477 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ℕ0𝑧 ∈ (1...𝑁)) → 𝑧 ∈ (1..^(1 + 𝑁)))
3121adantr 481 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ℕ0𝑧 ∈ (1...𝑁)) → 𝑁 ∈ ℤ)
32 fzosubel3 12936 . . . . . . . . . . . . . . . . 17 ((𝑧 ∈ (1..^(1 + 𝑁)) ∧ 𝑁 ∈ ℤ) → (𝑧 − 1) ∈ (0..^𝑁))
3330, 31, 32syl2anc 584 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ0𝑧 ∈ (1...𝑁)) → (𝑧 − 1) ∈ (0..^𝑁))
34 oveq1 7014 . . . . . . . . . . . . . . . . . 18 (𝑥 = (𝑧 − 1) → (𝑥 + 1) = ((𝑧 − 1) + 1))
3534eqeq2d 2803 . . . . . . . . . . . . . . . . 17 (𝑥 = (𝑧 − 1) → (𝑧 = (𝑥 + 1) ↔ 𝑧 = ((𝑧 − 1) + 1)))
3635adantl 482 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0𝑧 ∈ (1...𝑁)) ∧ 𝑥 = (𝑧 − 1)) → (𝑧 = (𝑥 + 1) ↔ 𝑧 = ((𝑧 − 1) + 1)))
37 elfzelz 12747 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ (1...𝑁) → 𝑧 ∈ ℤ)
3837zcnd 11926 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ (1...𝑁) → 𝑧 ∈ ℂ)
3938adantl 482 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ ℕ0𝑧 ∈ (1...𝑁)) → 𝑧 ∈ ℂ)
40 1cnd 10471 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ ℕ0𝑧 ∈ (1...𝑁)) → 1 ∈ ℂ)
4139, 40npcand 10838 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ℕ0𝑧 ∈ (1...𝑁)) → ((𝑧 − 1) + 1) = 𝑧)
4241eqcomd 2799 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ0𝑧 ∈ (1...𝑁)) → 𝑧 = ((𝑧 − 1) + 1))
4333, 36, 42rspcedvd 3561 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℕ0𝑧 ∈ (1...𝑁)) → ∃𝑥 ∈ (0..^𝑁)𝑧 = (𝑥 + 1))
44 fveq2 6530 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑥 + 1) → (𝐹𝑧) = (𝐹‘(𝑥 + 1)))
4544eqeq2d 2803 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑥 + 1) → (𝑦 = (𝐹𝑧) ↔ 𝑦 = (𝐹‘(𝑥 + 1))))
4645adantl 482 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℕ0𝑧 = (𝑥 + 1)) → (𝑦 = (𝐹𝑧) ↔ 𝑦 = (𝐹‘(𝑥 + 1))))
4720, 43, 46rexxfrd 5194 . . . . . . . . . . . . . 14 (𝑁 ∈ ℕ0 → (∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧) ↔ ∃𝑥 ∈ (0..^𝑁)𝑦 = (𝐹‘(𝑥 + 1))))
4847adantr 481 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ0 ∧ (♯‘𝐹) = 𝑁) → (∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧) ↔ ∃𝑥 ∈ (0..^𝑁)𝑦 = (𝐹‘(𝑥 + 1))))
49 oveq2 7015 . . . . . . . . . . . . . . . 16 ((♯‘𝐹) = 𝑁 → (0..^(♯‘𝐹)) = (0..^𝑁))
5049rexeqdv 3373 . . . . . . . . . . . . . . 15 ((♯‘𝐹) = 𝑁 → (∃𝑥 ∈ (0..^(♯‘𝐹))𝑦 = (𝐹‘(𝑥 + 1)) ↔ ∃𝑥 ∈ (0..^𝑁)𝑦 = (𝐹‘(𝑥 + 1))))
5150bibi2d 344 . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . 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 258 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0 ∧ (♯‘𝐹) = 𝑁) → (∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧) ↔ ∃𝑥 ∈ (0..^(♯‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))))
5419, 53syldan 591 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–onto→dom 𝐸) → (∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧) ↔ ∃𝑥 ∈ (0..^(♯‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))))
5554abbidv 2858 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–onto→dom 𝐸) → {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧)} = {𝑦 ∣ ∃𝑥 ∈ (0..^(♯‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))})
5655eqeq1d 2795 . . . . . . . . 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 250 . . . . . . . 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 254 . . . . . . 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 482 . . . . 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 2841 . 2 ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–onto→dom 𝐸) → ran 𝐺 = dom 𝐸)
64 dffo2 6454 . 2 (𝐺:(0..^(♯‘𝐹))–onto→dom 𝐸 ↔ (𝐺:(0..^(♯‘𝐹))⟶dom 𝐸 ∧ ran 𝐺 = dom 𝐸))
654, 63, 64sylanbrc 583 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 207  wa 396   = wceq 1520  wcel 2079  {cab 2773  wrex 3104  cmpt 5035  dom cdm 5435  ran crn 5436   Fn wfn 6212  wf 6213  ontowfo 6215  cfv 6217  (class class class)co 7007  cc 10370  0cc0 10372  1c1 10373   + caddc 10375  cmin 10706  0cn0 11734  cz 11818  ...cfz 12731  ..^cfzo 12872  chash 13528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-rep 5075  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214  ax-un 7310  ax-cnex 10428  ax-resscn 10429  ax-1cn 10430  ax-icn 10431  ax-addcl 10432  ax-addrcl 10433  ax-mulcl 10434  ax-mulrcl 10435  ax-mulcom 10436  ax-addass 10437  ax-mulass 10438  ax-distr 10439  ax-i2m1 10440  ax-1ne0 10441  ax-1rid 10442  ax-rnegex 10443  ax-rrecex 10444  ax-cnre 10445  ax-pre-lttri 10446  ax-pre-lttrn 10447  ax-pre-ltadd 10448  ax-pre-mulgt0 10449
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1079  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-nel 3089  df-ral 3108  df-rex 3109  df-reu 3110  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-pss 3871  df-nul 4207  df-if 4376  df-pw 4449  df-sn 4467  df-pr 4469  df-tp 4471  df-op 4473  df-uni 4740  df-int 4777  df-iun 4821  df-br 4957  df-opab 5019  df-mpt 5036  df-tr 5058  df-id 5340  df-eprel 5345  df-po 5354  df-so 5355  df-fr 5394  df-we 5396  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-pred 6015  df-ord 6061  df-on 6062  df-lim 6063  df-suc 6064  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-f1 6222  df-fo 6223  df-f1o 6224  df-fv 6225  df-riota 6968  df-ov 7010  df-oprab 7011  df-mpo 7012  df-om 7428  df-1st 7536  df-2nd 7537  df-wrecs 7789  df-recs 7851  df-rdg 7889  df-1o 7944  df-er 8130  df-en 8348  df-dom 8349  df-sdom 8350  df-fin 8351  df-card 9203  df-pnf 10512  df-mnf 10513  df-xr 10514  df-ltxr 10515  df-le 10516  df-sub 10708  df-neg 10709  df-nn 11476  df-n0 11735  df-z 11819  df-uz 12083  df-fz 12732  df-fzo 12873  df-hash 13529
This theorem is referenced by: (None)
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