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Theorem shftfn 15073
Description: Functionality and domain of a sequence shifted by 𝐴. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
Hypothesis
Ref Expression
shftfval.1 𝐹 ∈ V
Assertion
Ref Expression
shftfn ((𝐹 Fn 𝐵𝐴 ∈ ℂ) → (𝐹 shift 𝐴) Fn {𝑥 ∈ ℂ ∣ (𝑥𝐴) ∈ 𝐵})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐵

Proof of Theorem shftfn
Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relopabv 5819 . . . . 5 Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)}
21a1i 11 . . . 4 ((𝐹 Fn 𝐵𝐴 ∈ ℂ) → Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)})
3 fnfun 6652 . . . . . 6 (𝐹 Fn 𝐵 → Fun 𝐹)
43adantr 479 . . . . 5 ((𝐹 Fn 𝐵𝐴 ∈ ℂ) → Fun 𝐹)
5 funmo 6566 . . . . . . 7 (Fun 𝐹 → ∃*𝑤(𝑧𝐴)𝐹𝑤)
6 vex 3466 . . . . . . . . . 10 𝑧 ∈ V
7 vex 3466 . . . . . . . . . 10 𝑤 ∈ V
8 eleq1w 2809 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝑥 ∈ ℂ ↔ 𝑧 ∈ ℂ))
9 oveq1 7423 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑥𝐴) = (𝑧𝐴))
109breq1d 5155 . . . . . . . . . . 11 (𝑥 = 𝑧 → ((𝑥𝐴)𝐹𝑦 ↔ (𝑧𝐴)𝐹𝑦))
118, 10anbi12d 630 . . . . . . . . . 10 (𝑥 = 𝑧 → ((𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦) ↔ (𝑧 ∈ ℂ ∧ (𝑧𝐴)𝐹𝑦)))
12 breq2 5149 . . . . . . . . . . 11 (𝑦 = 𝑤 → ((𝑧𝐴)𝐹𝑦 ↔ (𝑧𝐴)𝐹𝑤))
1312anbi2d 628 . . . . . . . . . 10 (𝑦 = 𝑤 → ((𝑧 ∈ ℂ ∧ (𝑧𝐴)𝐹𝑦) ↔ (𝑧 ∈ ℂ ∧ (𝑧𝐴)𝐹𝑤)))
14 eqid 2726 . . . . . . . . . 10 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)}
156, 7, 11, 13, 14brab 5541 . . . . . . . . 9 (𝑧{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)}𝑤 ↔ (𝑧 ∈ ℂ ∧ (𝑧𝐴)𝐹𝑤))
1615simprbi 495 . . . . . . . 8 (𝑧{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)}𝑤 → (𝑧𝐴)𝐹𝑤)
1716moimi 2534 . . . . . . 7 (∃*𝑤(𝑧𝐴)𝐹𝑤 → ∃*𝑤 𝑧{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)}𝑤)
185, 17syl 17 . . . . . 6 (Fun 𝐹 → ∃*𝑤 𝑧{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)}𝑤)
1918alrimiv 1923 . . . . 5 (Fun 𝐹 → ∀𝑧∃*𝑤 𝑧{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)}𝑤)
204, 19syl 17 . . . 4 ((𝐹 Fn 𝐵𝐴 ∈ ℂ) → ∀𝑧∃*𝑤 𝑧{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)}𝑤)
21 dffun6 6559 . . . 4 (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)} ↔ (Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)} ∧ ∀𝑧∃*𝑤 𝑧{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)}𝑤))
222, 20, 21sylanbrc 581 . . 3 ((𝐹 Fn 𝐵𝐴 ∈ ℂ) → Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)})
23 shftfval.1 . . . . . 6 𝐹 ∈ V
2423shftfval 15070 . . . . 5 (𝐴 ∈ ℂ → (𝐹 shift 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)})
2524adantl 480 . . . 4 ((𝐹 Fn 𝐵𝐴 ∈ ℂ) → (𝐹 shift 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)})
2625funeqd 6573 . . 3 ((𝐹 Fn 𝐵𝐴 ∈ ℂ) → (Fun (𝐹 shift 𝐴) ↔ Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)}))
2722, 26mpbird 256 . 2 ((𝐹 Fn 𝐵𝐴 ∈ ℂ) → Fun (𝐹 shift 𝐴))
2823shftdm 15071 . . 3 (𝐴 ∈ ℂ → dom (𝐹 shift 𝐴) = {𝑥 ∈ ℂ ∣ (𝑥𝐴) ∈ dom 𝐹})
29 fndm 6655 . . . . 5 (𝐹 Fn 𝐵 → dom 𝐹 = 𝐵)
3029eleq2d 2812 . . . 4 (𝐹 Fn 𝐵 → ((𝑥𝐴) ∈ dom 𝐹 ↔ (𝑥𝐴) ∈ 𝐵))
3130rabbidv 3427 . . 3 (𝐹 Fn 𝐵 → {𝑥 ∈ ℂ ∣ (𝑥𝐴) ∈ dom 𝐹} = {𝑥 ∈ ℂ ∣ (𝑥𝐴) ∈ 𝐵})
3228, 31sylan9eqr 2788 . 2 ((𝐹 Fn 𝐵𝐴 ∈ ℂ) → dom (𝐹 shift 𝐴) = {𝑥 ∈ ℂ ∣ (𝑥𝐴) ∈ 𝐵})
33 df-fn 6549 . 2 ((𝐹 shift 𝐴) Fn {𝑥 ∈ ℂ ∣ (𝑥𝐴) ∈ 𝐵} ↔ (Fun (𝐹 shift 𝐴) ∧ dom (𝐹 shift 𝐴) = {𝑥 ∈ ℂ ∣ (𝑥𝐴) ∈ 𝐵}))
3427, 32, 33sylanbrc 581 1 ((𝐹 Fn 𝐵𝐴 ∈ ℂ) → (𝐹 shift 𝐴) Fn {𝑥 ∈ ℂ ∣ (𝑥𝐴) ∈ 𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wal 1532   = wceq 1534  wcel 2099  ∃*wmo 2527  {crab 3419  Vcvv 3462   class class class wbr 5145  {copab 5207  dom cdm 5674  Rel wrel 5679  Fun wfun 6540   Fn wfn 6541  (class class class)co 7416  cc 11147  cmin 11485   shift cshi 15066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738  ax-resscn 11206  ax-1cn 11207  ax-icn 11208  ax-addcl 11209  ax-addrcl 11210  ax-mulcl 11211  ax-mulrcl 11212  ax-mulcom 11213  ax-addass 11214  ax-mulass 11215  ax-distr 11216  ax-i2m1 11217  ax-1ne0 11218  ax-1rid 11219  ax-rnegex 11220  ax-rrecex 11221  ax-cnre 11222  ax-pre-lttri 11223  ax-pre-lttrn 11224  ax-pre-ltadd 11225
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-po 5586  df-so 5587  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-er 8726  df-en 8967  df-dom 8968  df-sdom 8969  df-pnf 11291  df-mnf 11292  df-ltxr 11294  df-sub 11487  df-shft 15067
This theorem is referenced by:  shftf  15079  seqshft  15085  uzmptshftfval  44057
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