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Theorem shftfn 10223
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 relopab 4552 . . . . 5 Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)}
21a1i 9 . . . 4 ((𝐹 Fn 𝐵𝐴 ∈ ℂ) → Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)})
3 fnfun 5097 . . . . . 6 (𝐹 Fn 𝐵 → Fun 𝐹)
43adantr 270 . . . . 5 ((𝐹 Fn 𝐵𝐴 ∈ ℂ) → Fun 𝐹)
5 funmo 5017 . . . . . . 7 (Fun 𝐹 → ∃*𝑤(𝑧𝐴)𝐹𝑤)
6 vex 2622 . . . . . . . . . 10 𝑧 ∈ V
7 vex 2622 . . . . . . . . . 10 𝑤 ∈ V
8 eleq1 2150 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝑥 ∈ ℂ ↔ 𝑧 ∈ ℂ))
9 oveq1 5641 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑥𝐴) = (𝑧𝐴))
109breq1d 3847 . . . . . . . . . . 11 (𝑥 = 𝑧 → ((𝑥𝐴)𝐹𝑦 ↔ (𝑧𝐴)𝐹𝑦))
118, 10anbi12d 457 . . . . . . . . . 10 (𝑥 = 𝑧 → ((𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦) ↔ (𝑧 ∈ ℂ ∧ (𝑧𝐴)𝐹𝑦)))
12 breq2 3841 . . . . . . . . . . 11 (𝑦 = 𝑤 → ((𝑧𝐴)𝐹𝑦 ↔ (𝑧𝐴)𝐹𝑤))
1312anbi2d 452 . . . . . . . . . 10 (𝑦 = 𝑤 → ((𝑧 ∈ ℂ ∧ (𝑧𝐴)𝐹𝑦) ↔ (𝑧 ∈ ℂ ∧ (𝑧𝐴)𝐹𝑤)))
14 eqid 2088 . . . . . . . . . 10 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)}
156, 7, 11, 13, 14brab 4090 . . . . . . . . 9 (𝑧{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)}𝑤 ↔ (𝑧 ∈ ℂ ∧ (𝑧𝐴)𝐹𝑤))
1615simprbi 269 . . . . . . . 8 (𝑧{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)}𝑤 → (𝑧𝐴)𝐹𝑤)
1716moimi 2013 . . . . . . 7 (∃*𝑤(𝑧𝐴)𝐹𝑤 → ∃*𝑤 𝑧{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)}𝑤)
185, 17syl 14 . . . . . 6 (Fun 𝐹 → ∃*𝑤 𝑧{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)}𝑤)
1918alrimiv 1802 . . . . 5 (Fun 𝐹 → ∀𝑧∃*𝑤 𝑧{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)}𝑤)
204, 19syl 14 . . . 4 ((𝐹 Fn 𝐵𝐴 ∈ ℂ) → ∀𝑧∃*𝑤 𝑧{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)}𝑤)
21 dffun6 5016 . . . 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 408 . . 3 ((𝐹 Fn 𝐵𝐴 ∈ ℂ) → Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)})
23 shftfval.1 . . . . . 6 𝐹 ∈ V
2423shftfval 10220 . . . . 5 (𝐴 ∈ ℂ → (𝐹 shift 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)})
2524adantl 271 . . . 4 ((𝐹 Fn 𝐵𝐴 ∈ ℂ) → (𝐹 shift 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)})
2625funeqd 5023 . . 3 ((𝐹 Fn 𝐵𝐴 ∈ ℂ) → (Fun (𝐹 shift 𝐴) ↔ Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)}))
2722, 26mpbird 165 . 2 ((𝐹 Fn 𝐵𝐴 ∈ ℂ) → Fun (𝐹 shift 𝐴))
2823shftdm 10221 . . 3 (𝐴 ∈ ℂ → dom (𝐹 shift 𝐴) = {𝑥 ∈ ℂ ∣ (𝑥𝐴) ∈ dom 𝐹})
29 fndm 5099 . . . . 5 (𝐹 Fn 𝐵 → dom 𝐹 = 𝐵)
3029eleq2d 2157 . . . 4 (𝐹 Fn 𝐵 → ((𝑥𝐴) ∈ dom 𝐹 ↔ (𝑥𝐴) ∈ 𝐵))
3130rabbidv 2608 . . 3 (𝐹 Fn 𝐵 → {𝑥 ∈ ℂ ∣ (𝑥𝐴) ∈ dom 𝐹} = {𝑥 ∈ ℂ ∣ (𝑥𝐴) ∈ 𝐵})
3228, 31sylan9eqr 2142 . 2 ((𝐹 Fn 𝐵𝐴 ∈ ℂ) → dom (𝐹 shift 𝐴) = {𝑥 ∈ ℂ ∣ (𝑥𝐴) ∈ 𝐵})
33 df-fn 5005 . 2 ((𝐹 shift 𝐴) Fn {𝑥 ∈ ℂ ∣ (𝑥𝐴) ∈ 𝐵} ↔ (Fun (𝐹 shift 𝐴) ∧ dom (𝐹 shift 𝐴) = {𝑥 ∈ ℂ ∣ (𝑥𝐴) ∈ 𝐵}))
3427, 32, 33sylanbrc 408 1 ((𝐹 Fn 𝐵𝐴 ∈ ℂ) → (𝐹 shift 𝐴) Fn {𝑥 ∈ ℂ ∣ (𝑥𝐴) ∈ 𝐵})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wal 1287   = wceq 1289  wcel 1438  ∃*wmo 1949  {crab 2363  Vcvv 2619   class class class wbr 3837  {copab 3890  dom cdm 4428  Rel wrel 4433  Fun wfun 4996   Fn wfn 4997  (class class class)co 5634  cc 7327  cmin 7632   shift cshi 10213
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-resscn 7416  ax-1cn 7417  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-addcom 7424  ax-addass 7426  ax-distr 7428  ax-i2m1 7429  ax-0id 7432  ax-rnegex 7433  ax-cnre 7435
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-sub 7634  df-shft 10214
This theorem is referenced by:  shftf  10229  seq3shft  10237
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