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Theorem shftfval 14263
Description: The value of the sequence shifter operation is a function on . 𝐴 is ordinarily an integer. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
Hypothesis
Ref Expression
shftfval.1 𝐹 ∈ V
Assertion
Ref Expression
shftfval (𝐴 ∈ ℂ → (𝐹 shift 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)})
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐹,𝑦

Proof of Theorem shftfval
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovex 7048 . . . . . . . . . 10 (𝑥𝐴) ∈ V
2 vex 3440 . . . . . . . . . 10 𝑦 ∈ V
31, 2breldm 5663 . . . . . . . . 9 ((𝑥𝐴)𝐹𝑦 → (𝑥𝐴) ∈ dom 𝐹)
4 npcan 10743 . . . . . . . . . . 11 ((𝑥 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((𝑥𝐴) + 𝐴) = 𝑥)
54eqcomd 2801 . . . . . . . . . 10 ((𝑥 ∈ ℂ ∧ 𝐴 ∈ ℂ) → 𝑥 = ((𝑥𝐴) + 𝐴))
65ancoms 459 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → 𝑥 = ((𝑥𝐴) + 𝐴))
7 oveq1 7023 . . . . . . . . . 10 (𝑤 = (𝑥𝐴) → (𝑤 + 𝐴) = ((𝑥𝐴) + 𝐴))
87rspceeqv 3577 . . . . . . . . 9 (((𝑥𝐴) ∈ dom 𝐹𝑥 = ((𝑥𝐴) + 𝐴)) → ∃𝑤 ∈ dom 𝐹 𝑥 = (𝑤 + 𝐴))
93, 6, 8syl2anr 596 . . . . . . . 8 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ (𝑥𝐴)𝐹𝑦) → ∃𝑤 ∈ dom 𝐹 𝑥 = (𝑤 + 𝐴))
10 vex 3440 . . . . . . . . 9 𝑥 ∈ V
11 eqeq1 2799 . . . . . . . . . 10 (𝑧 = 𝑥 → (𝑧 = (𝑤 + 𝐴) ↔ 𝑥 = (𝑤 + 𝐴)))
1211rexbidv 3260 . . . . . . . . 9 (𝑧 = 𝑥 → (∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴) ↔ ∃𝑤 ∈ dom 𝐹 𝑥 = (𝑤 + 𝐴)))
1310, 12elab 3605 . . . . . . . 8 (𝑥 ∈ {𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} ↔ ∃𝑤 ∈ dom 𝐹 𝑥 = (𝑤 + 𝐴))
149, 13sylibr 235 . . . . . . 7 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ (𝑥𝐴)𝐹𝑦) → 𝑥 ∈ {𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)})
151, 2brelrn 5694 . . . . . . . 8 ((𝑥𝐴)𝐹𝑦𝑦 ∈ ran 𝐹)
1615adantl 482 . . . . . . 7 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ (𝑥𝐴)𝐹𝑦) → 𝑦 ∈ ran 𝐹)
1714, 16jca 512 . . . . . 6 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ (𝑥𝐴)𝐹𝑦) → (𝑥 ∈ {𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} ∧ 𝑦 ∈ ran 𝐹))
1817expl 458 . . . . 5 (𝐴 ∈ ℂ → ((𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦) → (𝑥 ∈ {𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} ∧ 𝑦 ∈ ran 𝐹)))
1918ssopab2dv 5326 . . . 4 (𝐴 ∈ ℂ → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} ∧ 𝑦 ∈ ran 𝐹)})
20 df-xp 5449 . . . 4 ({𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} × ran 𝐹) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} ∧ 𝑦 ∈ ran 𝐹)}
2119, 20syl6sseqr 3939 . . 3 (𝐴 ∈ ℂ → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)} ⊆ ({𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} × ran 𝐹))
22 shftfval.1 . . . . . 6 𝐹 ∈ V
2322dmex 7472 . . . . 5 dom 𝐹 ∈ V
2423abrexex 7519 . . . 4 {𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} ∈ V
2522rnex 7473 . . . 4 ran 𝐹 ∈ V
2624, 25xpex 7333 . . 3 ({𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} × ran 𝐹) ∈ V
27 ssexg 5118 . . 3 (({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)} ⊆ ({𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} × ran 𝐹) ∧ ({𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} × ran 𝐹) ∈ V) → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)} ∈ V)
2821, 26, 27sylancl 586 . 2 (𝐴 ∈ ℂ → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)} ∈ V)
29 breq 4964 . . . . . 6 (𝑧 = 𝐹 → ((𝑥𝑤)𝑧𝑦 ↔ (𝑥𝑤)𝐹𝑦))
3029anbi2d 628 . . . . 5 (𝑧 = 𝐹 → ((𝑥 ∈ ℂ ∧ (𝑥𝑤)𝑧𝑦) ↔ (𝑥 ∈ ℂ ∧ (𝑥𝑤)𝐹𝑦)))
3130opabbidv 5028 . . . 4 (𝑧 = 𝐹 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝑤)𝑧𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝑤)𝐹𝑦)})
32 oveq2 7024 . . . . . . 7 (𝑤 = 𝐴 → (𝑥𝑤) = (𝑥𝐴))
3332breq1d 4972 . . . . . 6 (𝑤 = 𝐴 → ((𝑥𝑤)𝐹𝑦 ↔ (𝑥𝐴)𝐹𝑦))
3433anbi2d 628 . . . . 5 (𝑤 = 𝐴 → ((𝑥 ∈ ℂ ∧ (𝑥𝑤)𝐹𝑦) ↔ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)))
3534opabbidv 5028 . . . 4 (𝑤 = 𝐴 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝑤)𝐹𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)})
36 df-shft 14260 . . . 4 shift = (𝑧 ∈ V, 𝑤 ∈ ℂ ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝑤)𝑧𝑦)})
3731, 35, 36ovmpog 7165 . . 3 ((𝐹 ∈ V ∧ 𝐴 ∈ ℂ ∧ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)} ∈ V) → (𝐹 shift 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)})
3822, 37mp3an1 1440 . 2 ((𝐴 ∈ ℂ ∧ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)} ∈ V) → (𝐹 shift 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)})
3928, 38mpdan 683 1 (𝐴 ∈ ℂ → (𝐹 shift 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1522  wcel 2081  {cab 2775  wrex 3106  Vcvv 3437  wss 3859   class class class wbr 4962  {copab 5024   × cxp 5441  dom cdm 5443  ran crn 5444  (class class class)co 7016  cc 10381   + caddc 10386  cmin 10717   shift cshi 14259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-po 5362  df-so 5363  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-er 8139  df-en 8358  df-dom 8359  df-sdom 8360  df-pnf 10523  df-mnf 10524  df-ltxr 10526  df-sub 10719  df-shft 14260
This theorem is referenced by:  shftdm  14264  shftfib  14265  shftfn  14266  2shfti  14273  shftidt2  14274
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