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Theorem fargshiftfo 47316
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 6834 . . 3 (𝐹:(1...𝑁)–onto→dom 𝐸𝐹:(1...𝑁)⟶dom 𝐸)
2 fargshift.g . . . 4 𝐺 = (𝑥 ∈ (0..^(♯‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))
32fargshiftf 47314 . . 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 5980 . . 3 ran 𝐺 = {𝑦 ∣ ∃𝑥 ∈ (0..^(♯‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))}
6 fofn 6836 . . . . . 6 (𝐹:(1...𝑁)–onto→dom 𝐸𝐹 Fn (1...𝑁))
7 fnrnfv 6981 . . . . . 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 6579 . . . . . . 7 (𝐹:(1...𝑁)–onto→dom 𝐸 ↔ (𝐹 Fn (1...𝑁) ∧ ran 𝐹 = dom 𝐸))
1110biimpi 216 . . . . . 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 2744 . . . . . . . . 9 (ran 𝐹 = dom 𝐸 → (ran 𝐹 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧)} ↔ dom 𝐸 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧)}))
14 eqcom 2747 . . . . . . . . 9 (dom 𝐸 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧)} ↔ {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧)} = dom 𝐸)
1513, 14bitrdi 287 . . . . . . . 8 (ran 𝐹 = dom 𝐸 → (ran 𝐹 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧)} ↔ {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧)} = dom 𝐸))
16 ffn 6747 . . . . . . . . . . . . . 14 (𝐹:(1...𝑁)⟶dom 𝐸𝐹 Fn (1...𝑁))
17 fseq1hash 14425 . . . . . . . . . . . . . 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 13786 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℕ0𝑥 ∈ (0..^𝑁)) → (𝑥 + 1) ∈ (1...𝑁))
21 nn0z 12664 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0𝑁 ∈ ℤ)
22 fzval3 13785 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℤ → (1...𝑁) = (1..^(𝑁 + 1)))
2321, 22syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → (1...𝑁) = (1..^(𝑁 + 1)))
24 nn0cn 12563 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0𝑁 ∈ ℂ)
25 1cnd 11285 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0 → 1 ∈ ℂ)
2624, 25addcomd 11492 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → (𝑁 + 1) = (1 + 𝑁))
2726oveq2d 7464 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → (1..^(𝑁 + 1)) = (1..^(1 + 𝑁)))
2823, 27eqtrd 2780 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ0 → (1...𝑁) = (1..^(1 + 𝑁)))
2928eleq2d 2830 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ ℕ0 → (𝑧 ∈ (1...𝑁) ↔ 𝑧 ∈ (1..^(1 + 𝑁))))
3029biimpa 476 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ℕ0𝑧 ∈ (1...𝑁)) → 𝑧 ∈ (1..^(1 + 𝑁)))
3121adantr 480 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ℕ0𝑧 ∈ (1...𝑁)) → 𝑁 ∈ ℤ)
32 fzosubel3 13777 . . . . . . . . . . . . . . . . 17 ((𝑧 ∈ (1..^(1 + 𝑁)) ∧ 𝑁 ∈ ℤ) → (𝑧 − 1) ∈ (0..^𝑁))
3330, 31, 32syl2anc 583 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ0𝑧 ∈ (1...𝑁)) → (𝑧 − 1) ∈ (0..^𝑁))
34 oveq1 7455 . . . . . . . . . . . . . . . . . 18 (𝑥 = (𝑧 − 1) → (𝑥 + 1) = ((𝑧 − 1) + 1))
3534eqeq2d 2751 . . . . . . . . . . . . . . . . 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 13584 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ (1...𝑁) → 𝑧 ∈ ℤ)
3837zcnd 12748 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ (1...𝑁) → 𝑧 ∈ ℂ)
3938adantl 481 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ ℕ0𝑧 ∈ (1...𝑁)) → 𝑧 ∈ ℂ)
40 1cnd 11285 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ ℕ0𝑧 ∈ (1...𝑁)) → 1 ∈ ℂ)
4139, 40npcand 11651 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ℕ0𝑧 ∈ (1...𝑁)) → ((𝑧 − 1) + 1) = 𝑧)
4241eqcomd 2746 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ0𝑧 ∈ (1...𝑁)) → 𝑧 = ((𝑧 − 1) + 1))
4333, 36, 42rspcedvd 3637 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℕ0𝑧 ∈ (1...𝑁)) → ∃𝑥 ∈ (0..^𝑁)𝑧 = (𝑥 + 1))
44 fveq2 6920 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑥 + 1) → (𝐹𝑧) = (𝐹‘(𝑥 + 1)))
4544eqeq2d 2751 . . . . . . . . . . . . . . . 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 5427 . . . . . . . . . . . . . 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 7456 . . . . . . . . . . . . . . . 16 ((♯‘𝐹) = 𝑁 → (0..^(♯‘𝐹)) = (0..^𝑁))
5049rexeqdv 3335 . . . . . . . . . . . . . . 15 ((♯‘𝐹) = 𝑁 → (∃𝑥 ∈ (0..^(♯‘𝐹))𝑦 = (𝐹‘(𝑥 + 1)) ↔ ∃𝑥 ∈ (0..^𝑁)𝑦 = (𝐹‘(𝑥 + 1))))
5150bibi2d 342 . . . . . . . . . . . . . 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 257 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0 ∧ (♯‘𝐹) = 𝑁) → (∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧) ↔ ∃𝑥 ∈ (0..^(♯‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))))
5419, 53syldan 590 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–onto→dom 𝐸) → (∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧) ↔ ∃𝑥 ∈ (0..^(♯‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))))
5554abbidv 2811 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–onto→dom 𝐸) → {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧)} = {𝑦 ∣ ∃𝑥 ∈ (0..^(♯‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))})
5655eqeq1d 2742 . . . . . . . . 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 249 . . . . . . . 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, 57biimtrdi 253 . . . . . . 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, 62eqtrid 2792 . 2 ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–onto→dom 𝐸) → ran 𝐺 = dom 𝐸)
64 dffo2 6838 . 2 (𝐺:(0..^(♯‘𝐹))–onto→dom 𝐸 ↔ (𝐺:(0..^(♯‘𝐹))⟶dom 𝐸 ∧ ran 𝐺 = dom 𝐸))
654, 63, 64sylanbrc 582 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 206  wa 395   = wceq 1537  wcel 2108  {cab 2717  wrex 3076  cmpt 5249  dom cdm 5700  ran crn 5701   Fn wfn 6568  wf 6569  ontowfo 6571  cfv 6573  (class class class)co 7448  cc 11182  0cc0 11184  1c1 11185   + caddc 11187  cmin 11520  0cn0 12553  cz 12639  ...cfz 13567  ..^cfzo 13711  chash 14379
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-n0 12554  df-z 12640  df-uz 12904  df-fz 13568  df-fzo 13712  df-hash 14380
This theorem is referenced by: (None)
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