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Theorem fargshiftfo 26166
 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 6028 . . 3 (𝐹:(1...𝑁)–onto→dom 𝐸𝐹:(1...𝑁)⟶dom 𝐸)
2 fargshift.g . . . 4 𝐺 = (𝑥 ∈ (0..^(#‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))
32fargshiftf 26164 . . 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 5292 . . 3 ran 𝐺 = {𝑦 ∣ ∃𝑥 ∈ (0..^(#‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))}
6 fofn 6030 . . . . . 6 (𝐹:(1...𝑁)–onto→dom 𝐸𝐹 Fn (1...𝑁))
7 fnrnfv 6152 . . . . . 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 5810 . . . . . . 7 (𝐹:(1...𝑁)–onto→dom 𝐸 ↔ (𝐹 Fn (1...𝑁) ∧ ran 𝐹 = dom 𝐸))
1110biimpi 205 . . . . . 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 2614 . . . . . . . . 9 (ran 𝐹 = dom 𝐸 → (ran 𝐹 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧)} ↔ dom 𝐸 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧)}))
14 eqcom 2617 . . . . . . . . 9 (dom 𝐸 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧)} ↔ {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧)} = dom 𝐸)
1513, 14syl6bb 275 . . . . . . . 8 (ran 𝐹 = dom 𝐸 → (ran 𝐹 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧)} ↔ {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧)} = dom 𝐸))
16 ffn 5958 . . . . . . . . . . . . . 14 (𝐹:(1...𝑁)⟶dom 𝐸𝐹 Fn (1...𝑁))
17 fseq1hash 13026 . . . . . . . . . . . . . 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 fargshiftlem 26162 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℕ0𝑥 ∈ (0..^𝑁)) → (𝑥 + 1) ∈ (1...𝑁))
21 nn0z 11277 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0𝑁 ∈ ℤ)
22 fzval3 12404 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℤ → (1...𝑁) = (1..^(𝑁 + 1)))
2321, 22syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → (1...𝑁) = (1..^(𝑁 + 1)))
24 nn0cn 11179 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0𝑁 ∈ ℂ)
25 1cnd 9935 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0 → 1 ∈ ℂ)
2624, 25addcomd 10117 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → (𝑁 + 1) = (1 + 𝑁))
2726oveq2d 6565 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → (1..^(𝑁 + 1)) = (1..^(1 + 𝑁)))
2823, 27eqtrd 2644 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ0 → (1...𝑁) = (1..^(1 + 𝑁)))
2928eleq2d 2673 . . . . . . . . . . . . . . . . . 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 12396 . . . . . . . . . . . . . . . . 17 ((𝑧 ∈ (1..^(1 + 𝑁)) ∧ 𝑁 ∈ ℤ) → (𝑧 − 1) ∈ (0..^𝑁))
3330, 31, 32syl2anc 691 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ0𝑧 ∈ (1...𝑁)) → (𝑧 − 1) ∈ (0..^𝑁))
34 oveq1 6556 . . . . . . . . . . . . . . . . . 18 (𝑥 = (𝑧 − 1) → (𝑥 + 1) = ((𝑧 − 1) + 1))
3534eqeq2d 2620 . . . . . . . . . . . . . . . . 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 12213 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ (1...𝑁) → 𝑧 ∈ ℤ)
3837zcnd 11359 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ (1...𝑁) → 𝑧 ∈ ℂ)
3938adantl 481 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ ℕ0𝑧 ∈ (1...𝑁)) → 𝑧 ∈ ℂ)
40 1cnd 9935 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ ℕ0𝑧 ∈ (1...𝑁)) → 1 ∈ ℂ)
4139, 40npcand 10275 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ℕ0𝑧 ∈ (1...𝑁)) → ((𝑧 − 1) + 1) = 𝑧)
4241eqcomd 2616 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ0𝑧 ∈ (1...𝑁)) → 𝑧 = ((𝑧 − 1) + 1))
4333, 36, 42rspcedvd 3289 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℕ0𝑧 ∈ (1...𝑁)) → ∃𝑥 ∈ (0..^𝑁)𝑧 = (𝑥 + 1))
44 fveq2 6103 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑥 + 1) → (𝐹𝑧) = (𝐹‘(𝑥 + 1)))
4544eqeq2d 2620 . . . . . . . . . . . . . . . 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 4807 . . . . . . . . . . . . . 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 6557 . . . . . . . . . . . . . . . 16 ((#‘𝐹) = 𝑁 → (0..^(#‘𝐹)) = (0..^𝑁))
5049rexeqdv 3122 . . . . . . . . . . . . . . 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 246 . . . . . . . . . . . 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 2728 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–onto→dom 𝐸) → {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹𝑧)} = {𝑦 ∣ ∃𝑥 ∈ (0..^(#‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))})
5655eqeq1d 2612 . . . . . . . . 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 238 . . . . . . . 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 242 . . . . . . 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 84 . . . . . 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 37 . . . 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 2656 . 2 ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–onto→dom 𝐸) → ran 𝐺 = dom 𝐸)
64 dffo2 6032 . 2 (𝐺:(0..^(#‘𝐹))–onto→dom 𝐸 ↔ (𝐺:(0..^(#‘𝐹))⟶dom 𝐸 ∧ ran 𝐺 = dom 𝐸))
654, 63, 64sylanbrc 695 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 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  ∃wrex 2897   ↦ cmpt 4643  dom cdm 5038  ran crn 5039   Fn wfn 5799  ⟶wf 5800  –onto→wfo 5802  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  0cc0 9815  1c1 9816   + caddc 9818   − cmin 10145  ℕ0cn0 11169  ℤcz 11254  ...cfz 12197  ..^cfzo 12334  #chash 12979 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980 This theorem is referenced by:  eupatrl  26495
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