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Theorem fargshiftfo 44376
 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 6581 . . 3 (𝐹:(1...𝑁)–onto→dom 𝐸𝐹:(1...𝑁)⟶dom 𝐸)
2 fargshift.g . . . 4 𝐺 = (𝑥 ∈ (0..^(♯‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))
32fargshiftf 44374 . . 3 ((𝑁 ∈ ℕ0𝐹:(1...𝑁)⟶dom 𝐸) → 𝐺:(0..^(♯‘𝐹))⟶dom 𝐸)
41, 3sylan2 595 . 2 ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–onto→dom 𝐸) → 𝐺:(0..^(♯‘𝐹))⟶dom 𝐸)
52rnmpt 5801 . . 3 ran 𝐺 = {𝑦 ∣ ∃𝑥 ∈ (0..^(♯‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))}
6 fofn 6583 . . . . . 6 (𝐹:(1...𝑁)–onto→dom 𝐸𝐹 Fn (1...𝑁))
7 fnrnfv 6718 . . . . . 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 485 . . . 4 ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–onto→dom 𝐸) → ran 𝐹 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧)})
10 df-fo 6346 . . . . . . 7 (𝐹:(1...𝑁)–onto→dom 𝐸 ↔ (𝐹 Fn (1...𝑁) ∧ ran 𝐹 = dom 𝐸))
1110biimpi 219 . . . . . 6 (𝐹:(1...𝑁)–onto→dom 𝐸 → (𝐹 Fn (1...𝑁) ∧ ran 𝐹 = dom 𝐸))
1211adantl 485 . . . . 5 ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–onto→dom 𝐸) → (𝐹 Fn (1...𝑁) ∧ ran 𝐹 = dom 𝐸))
13 eqeq1 2762 . . . . . . . . 9 (ran 𝐹 = dom 𝐸 → (ran 𝐹 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧)} ↔ dom 𝐸 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧)}))
14 eqcom 2765 . . . . . . . . 9 (dom 𝐸 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧)} ↔ {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧)} = dom 𝐸)
1513, 14bitrdi 290 . . . . . . . 8 (ran 𝐹 = dom 𝐸 → (ran 𝐹 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧)} ↔ {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧)} = dom 𝐸))
16 ffn 6503 . . . . . . . . . . . . . 14 (𝐹:(1...𝑁)⟶dom 𝐸𝐹 Fn (1...𝑁))
17 fseq1hash 13800 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℕ0𝐹 Fn (1...𝑁)) → (♯‘𝐹) = 𝑁)
1816, 17sylan2 595 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ0𝐹:(1...𝑁)⟶dom 𝐸) → (♯‘𝐹) = 𝑁)
191, 18sylan2 595 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–onto→dom 𝐸) → (♯‘𝐹) = 𝑁)
20 fz0add1fz1 13169 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℕ0𝑥 ∈ (0..^𝑁)) → (𝑥 + 1) ∈ (1...𝑁))
21 nn0z 12057 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0𝑁 ∈ ℤ)
22 fzval3 13168 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℤ → (1...𝑁) = (1..^(𝑁 + 1)))
2321, 22syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → (1...𝑁) = (1..^(𝑁 + 1)))
24 nn0cn 11957 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0𝑁 ∈ ℂ)
25 1cnd 10687 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0 → 1 ∈ ℂ)
2624, 25addcomd 10893 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → (𝑁 + 1) = (1 + 𝑁))
2726oveq2d 7172 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → (1..^(𝑁 + 1)) = (1..^(1 + 𝑁)))
2823, 27eqtrd 2793 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ0 → (1...𝑁) = (1..^(1 + 𝑁)))
2928eleq2d 2837 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ ℕ0 → (𝑧 ∈ (1...𝑁) ↔ 𝑧 ∈ (1..^(1 + 𝑁))))
3029biimpa 480 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ℕ0𝑧 ∈ (1...𝑁)) → 𝑧 ∈ (1..^(1 + 𝑁)))
3121adantr 484 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ℕ0𝑧 ∈ (1...𝑁)) → 𝑁 ∈ ℤ)
32 fzosubel3 13160 . . . . . . . . . . . . . . . . 17 ((𝑧 ∈ (1..^(1 + 𝑁)) ∧ 𝑁 ∈ ℤ) → (𝑧 − 1) ∈ (0..^𝑁))
3330, 31, 32syl2anc 587 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ0𝑧 ∈ (1...𝑁)) → (𝑧 − 1) ∈ (0..^𝑁))
34 oveq1 7163 . . . . . . . . . . . . . . . . . 18 (𝑥 = (𝑧 − 1) → (𝑥 + 1) = ((𝑧 − 1) + 1))
3534eqeq2d 2769 . . . . . . . . . . . . . . . . 17 (𝑥 = (𝑧 − 1) → (𝑧 = (𝑥 + 1) ↔ 𝑧 = ((𝑧 − 1) + 1)))
3635adantl 485 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0𝑧 ∈ (1...𝑁)) ∧ 𝑥 = (𝑧 − 1)) → (𝑧 = (𝑥 + 1) ↔ 𝑧 = ((𝑧 − 1) + 1)))
37 elfzelz 12969 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ (1...𝑁) → 𝑧 ∈ ℤ)
3837zcnd 12140 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ (1...𝑁) → 𝑧 ∈ ℂ)
3938adantl 485 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ ℕ0𝑧 ∈ (1...𝑁)) → 𝑧 ∈ ℂ)
40 1cnd 10687 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ ℕ0𝑧 ∈ (1...𝑁)) → 1 ∈ ℂ)
4139, 40npcand 11052 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ℕ0𝑧 ∈ (1...𝑁)) → ((𝑧 − 1) + 1) = 𝑧)
4241eqcomd 2764 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ0𝑧 ∈ (1...𝑁)) → 𝑧 = ((𝑧 − 1) + 1))
4333, 36, 42rspcedvd 3546 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℕ0𝑧 ∈ (1...𝑁)) → ∃𝑥 ∈ (0..^𝑁)𝑧 = (𝑥 + 1))
44 fveq2 6663 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑥 + 1) → (𝐹𝑧) = (𝐹‘(𝑥 + 1)))
4544eqeq2d 2769 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑥 + 1) → (𝑦 = (𝐹𝑧) ↔ 𝑦 = (𝐹‘(𝑥 + 1))))
4645adantl 485 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℕ0𝑧 = (𝑥 + 1)) → (𝑦 = (𝐹𝑧) ↔ 𝑦 = (𝐹‘(𝑥 + 1))))
4720, 43, 46rexxfrd 5282 . . . . . . . . . . . . . 14 (𝑁 ∈ ℕ0 → (∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧) ↔ ∃𝑥 ∈ (0..^𝑁)𝑦 = (𝐹‘(𝑥 + 1))))
4847adantr 484 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ0 ∧ (♯‘𝐹) = 𝑁) → (∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧) ↔ ∃𝑥 ∈ (0..^𝑁)𝑦 = (𝐹‘(𝑥 + 1))))
49 oveq2 7164 . . . . . . . . . . . . . . . 16 ((♯‘𝐹) = 𝑁 → (0..^(♯‘𝐹)) = (0..^𝑁))
5049rexeqdv 3330 . . . . . . . . . . . . . . 15 ((♯‘𝐹) = 𝑁 → (∃𝑥 ∈ (0..^(♯‘𝐹))𝑦 = (𝐹‘(𝑥 + 1)) ↔ ∃𝑥 ∈ (0..^𝑁)𝑦 = (𝐹‘(𝑥 + 1))))
5150bibi2d 346 . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . 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 260 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0 ∧ (♯‘𝐹) = 𝑁) → (∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧) ↔ ∃𝑥 ∈ (0..^(♯‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))))
5419, 53syldan 594 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–onto→dom 𝐸) → (∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧) ↔ ∃𝑥 ∈ (0..^(♯‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))))
5554abbidv 2822 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–onto→dom 𝐸) → {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧)} = {𝑦 ∣ ∃𝑥 ∈ (0..^(♯‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))})
5655eqeq1d 2760 . . . . . . . . 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 252 . . . . . . . 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 256 . . . . . . 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 485 . . . . 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 2805 . 2 ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–onto→dom 𝐸) → ran 𝐺 = dom 𝐸)
64 dffo2 6585 . 2 (𝐺:(0..^(♯‘𝐹))–onto→dom 𝐸 ↔ (𝐺:(0..^(♯‘𝐹))⟶dom 𝐸 ∧ ran 𝐺 = dom 𝐸))
654, 63, 64sylanbrc 586 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 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2735  ∃wrex 3071   ↦ cmpt 5116  dom cdm 5528  ran crn 5529   Fn wfn 6335  ⟶wf 6336  –onto→wfo 6338  ‘cfv 6340  (class class class)co 7156  ℂcc 10586  0cc0 10588  1c1 10589   + caddc 10591   − cmin 10921  ℕ0cn0 11947  ℤcz 12033  ...cfz 12952  ..^cfzo 13095  ♯chash 13753 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465  ax-cnex 10644  ax-resscn 10645  ax-1cn 10646  ax-icn 10647  ax-addcl 10648  ax-addrcl 10649  ax-mulcl 10650  ax-mulrcl 10651  ax-mulcom 10652  ax-addass 10653  ax-mulass 10654  ax-distr 10655  ax-i2m1 10656  ax-1ne0 10657  ax-1rid 10658  ax-rnegex 10659  ax-rrecex 10660  ax-cnre 10661  ax-pre-lttri 10662  ax-pre-lttrn 10663  ax-pre-ltadd 10664  ax-pre-mulgt0 10665 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7586  df-1st 7699  df-2nd 7700  df-wrecs 7963  df-recs 8024  df-rdg 8062  df-er 8305  df-en 8541  df-dom 8542  df-sdom 8543  df-fin 8544  df-card 9414  df-pnf 10728  df-mnf 10729  df-xr 10730  df-ltxr 10731  df-le 10732  df-sub 10923  df-neg 10924  df-nn 11688  df-n0 11948  df-z 12034  df-uz 12296  df-fz 12953  df-fzo 13096  df-hash 13754 This theorem is referenced by: (None)
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