ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  shftfn GIF version

Theorem shftfn 10817
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 4750 . . . . 5 Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)}
21a1i 9 . . . 4 ((𝐹 Fn 𝐵𝐴 ∈ ℂ) → Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)})
3 fnfun 5309 . . . . . 6 (𝐹 Fn 𝐵 → Fun 𝐹)
43adantr 276 . . . . 5 ((𝐹 Fn 𝐵𝐴 ∈ ℂ) → Fun 𝐹)
5 funmo 5227 . . . . . . 7 (Fun 𝐹 → ∃*𝑤(𝑧𝐴)𝐹𝑤)
6 vex 2740 . . . . . . . . . 10 𝑧 ∈ V
7 vex 2740 . . . . . . . . . 10 𝑤 ∈ V
8 eleq1 2240 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝑥 ∈ ℂ ↔ 𝑧 ∈ ℂ))
9 oveq1 5876 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑥𝐴) = (𝑧𝐴))
109breq1d 4010 . . . . . . . . . . 11 (𝑥 = 𝑧 → ((𝑥𝐴)𝐹𝑦 ↔ (𝑧𝐴)𝐹𝑦))
118, 10anbi12d 473 . . . . . . . . . 10 (𝑥 = 𝑧 → ((𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦) ↔ (𝑧 ∈ ℂ ∧ (𝑧𝐴)𝐹𝑦)))
12 breq2 4004 . . . . . . . . . . 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 4269 . . . . . . . . 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 5226 . . . 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 10814 . . . . 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 5234 . . 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 10815 . . 3 (𝐴 ∈ ℂ → dom (𝐹 shift 𝐴) = {𝑥 ∈ ℂ ∣ (𝑥𝐴) ∈ dom 𝐹})
29 fndm 5311 . . . . 5 (𝐹 Fn 𝐵 → dom 𝐹 = 𝐵)
3029eleq2d 2247 . . . 4 (𝐹 Fn 𝐵 → ((𝑥𝐴) ∈ dom 𝐹 ↔ (𝑥𝐴) ∈ 𝐵))
3130rabbidv 2726 . . 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 5215 . 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 2737   class class class wbr 4000  {copab 4060  dom cdm 4623  Rel wrel 4628  Fun wfun 5206   Fn wfn 5207  (class class class)co 5869  cc 7800  cmin 8118   shift cshi 10807
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 4115  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-resscn 7894  ax-1cn 7895  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-addcom 7902  ax-addass 7904  ax-distr 7906  ax-i2m1 7907  ax-0id 7910  ax-rnegex 7911  ax-cnre 7913
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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-sub 8120  df-shft 10808
This theorem is referenced by:  shftf  10823  seq3shft  10831
  Copyright terms: Public domain W3C validator