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Theorem shftfn 10833
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 4754 . . . . 5 Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)}
21a1i 9 . . . 4 ((𝐹 Fn 𝐵𝐴 ∈ ℂ) → Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)})
3 fnfun 5314 . . . . . 6 (𝐹 Fn 𝐵 → Fun 𝐹)
43adantr 276 . . . . 5 ((𝐹 Fn 𝐵𝐴 ∈ ℂ) → Fun 𝐹)
5 funmo 5232 . . . . . . 7 (Fun 𝐹 → ∃*𝑤(𝑧𝐴)𝐹𝑤)
6 vex 2741 . . . . . . . . . 10 𝑧 ∈ V
7 vex 2741 . . . . . . . . . 10 𝑤 ∈ V
8 eleq1 2240 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝑥 ∈ ℂ ↔ 𝑧 ∈ ℂ))
9 oveq1 5882 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑥𝐴) = (𝑧𝐴))
109breq1d 4014 . . . . . . . . . . 11 (𝑥 = 𝑧 → ((𝑥𝐴)𝐹𝑦 ↔ (𝑧𝐴)𝐹𝑦))
118, 10anbi12d 473 . . . . . . . . . 10 (𝑥 = 𝑧 → ((𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦) ↔ (𝑧 ∈ ℂ ∧ (𝑧𝐴)𝐹𝑦)))
12 breq2 4008 . . . . . . . . . . 11 (𝑦 = 𝑤 → ((𝑧𝐴)𝐹𝑦 ↔ (𝑧𝐴)𝐹𝑤))
1312anbi2d 464 . . . . . . . . . 10 (𝑦 = 𝑤 → ((𝑧 ∈ ℂ ∧ (𝑧𝐴)𝐹𝑦) ↔ (𝑧 ∈ ℂ ∧ (𝑧𝐴)𝐹𝑤)))
14 eqid 2177 . . . . . . . . . 10 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)}
156, 7, 11, 13, 14brab 4273 . . . . . . . . 9 (𝑧{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)}𝑤 ↔ (𝑧 ∈ ℂ ∧ (𝑧𝐴)𝐹𝑤))
1615simprbi 275 . . . . . . . 8 (𝑧{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)}𝑤 → (𝑧𝐴)𝐹𝑤)
1716moimi 2091 . . . . . . 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 1874 . . . . 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 5231 . . . 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 417 . . 3 ((𝐹 Fn 𝐵𝐴 ∈ ℂ) → Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)})
23 shftfval.1 . . . . . 6 𝐹 ∈ V
2423shftfval 10830 . . . . 5 (𝐴 ∈ ℂ → (𝐹 shift 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)})
2524adantl 277 . . . 4 ((𝐹 Fn 𝐵𝐴 ∈ ℂ) → (𝐹 shift 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)})
2625funeqd 5239 . . 3 ((𝐹 Fn 𝐵𝐴 ∈ ℂ) → (Fun (𝐹 shift 𝐴) ↔ Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)}))
2722, 26mpbird 167 . 2 ((𝐹 Fn 𝐵𝐴 ∈ ℂ) → Fun (𝐹 shift 𝐴))
2823shftdm 10831 . . 3 (𝐴 ∈ ℂ → dom (𝐹 shift 𝐴) = {𝑥 ∈ ℂ ∣ (𝑥𝐴) ∈ dom 𝐹})
29 fndm 5316 . . . . 5 (𝐹 Fn 𝐵 → dom 𝐹 = 𝐵)
3029eleq2d 2247 . . . 4 (𝐹 Fn 𝐵 → ((𝑥𝐴) ∈ dom 𝐹 ↔ (𝑥𝐴) ∈ 𝐵))
3130rabbidv 2727 . . 3 (𝐹 Fn 𝐵 → {𝑥 ∈ ℂ ∣ (𝑥𝐴) ∈ dom 𝐹} = {𝑥 ∈ ℂ ∣ (𝑥𝐴) ∈ 𝐵})
3228, 31sylan9eqr 2232 . 2 ((𝐹 Fn 𝐵𝐴 ∈ ℂ) → dom (𝐹 shift 𝐴) = {𝑥 ∈ ℂ ∣ (𝑥𝐴) ∈ 𝐵})
33 df-fn 5220 . 2 ((𝐹 shift 𝐴) Fn {𝑥 ∈ ℂ ∣ (𝑥𝐴) ∈ 𝐵} ↔ (Fun (𝐹 shift 𝐴) ∧ dom (𝐹 shift 𝐴) = {𝑥 ∈ ℂ ∣ (𝑥𝐴) ∈ 𝐵}))
3427, 32, 33sylanbrc 417 1 ((𝐹 Fn 𝐵𝐴 ∈ ℂ) → (𝐹 shift 𝐴) Fn {𝑥 ∈ ℂ ∣ (𝑥𝐴) ∈ 𝐵})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wal 1351   = wceq 1353  ∃*wmo 2027  wcel 2148  {crab 2459  Vcvv 2738   class class class wbr 4004  {copab 4064  dom cdm 4627  Rel wrel 4632  Fun wfun 5211   Fn wfn 5212  (class class class)co 5875  cc 7809  cmin 8128   shift cshi 10823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-resscn 7903  ax-1cn 7904  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-addcom 7911  ax-addass 7913  ax-distr 7915  ax-i2m1 7916  ax-0id 7919  ax-rnegex 7920  ax-cnre 7922
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-sub 8130  df-shft 10824
This theorem is referenced by:  shftf  10839  seq3shft  10847
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