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Theorem shftfval 15049
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 7453 . . . . . . . . . 10 (𝑥𝐴) ∈ V
2 vex 3475 . . . . . . . . . 10 𝑦 ∈ V
31, 2breldm 5911 . . . . . . . . 9 ((𝑥𝐴)𝐹𝑦 → (𝑥𝐴) ∈ dom 𝐹)
4 npcan 11499 . . . . . . . . . . 11 ((𝑥 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((𝑥𝐴) + 𝐴) = 𝑥)
54eqcomd 2734 . . . . . . . . . 10 ((𝑥 ∈ ℂ ∧ 𝐴 ∈ ℂ) → 𝑥 = ((𝑥𝐴) + 𝐴))
65ancoms 458 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → 𝑥 = ((𝑥𝐴) + 𝐴))
7 oveq1 7427 . . . . . . . . . 10 (𝑤 = (𝑥𝐴) → (𝑤 + 𝐴) = ((𝑥𝐴) + 𝐴))
87rspceeqv 3631 . . . . . . . . 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 3475 . . . . . . . . 9 𝑥 ∈ V
11 eqeq1 2732 . . . . . . . . . 10 (𝑧 = 𝑥 → (𝑧 = (𝑤 + 𝐴) ↔ 𝑥 = (𝑤 + 𝐴)))
1211rexbidv 3175 . . . . . . . . 9 (𝑧 = 𝑥 → (∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴) ↔ ∃𝑤 ∈ dom 𝐹 𝑥 = (𝑤 + 𝐴)))
1310, 12elab 3667 . . . . . . . 8 (𝑥 ∈ {𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} ↔ ∃𝑤 ∈ dom 𝐹 𝑥 = (𝑤 + 𝐴))
149, 13sylibr 233 . . . . . . 7 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ (𝑥𝐴)𝐹𝑦) → 𝑥 ∈ {𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)})
151, 2brelrn 5944 . . . . . . . 8 ((𝑥𝐴)𝐹𝑦𝑦 ∈ ran 𝐹)
1615adantl 481 . . . . . . 7 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ (𝑥𝐴)𝐹𝑦) → 𝑦 ∈ ran 𝐹)
1714, 16jca 511 . . . . . 6 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ (𝑥𝐴)𝐹𝑦) → (𝑥 ∈ {𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} ∧ 𝑦 ∈ ran 𝐹))
1817expl 457 . . . . 5 (𝐴 ∈ ℂ → ((𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦) → (𝑥 ∈ {𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} ∧ 𝑦 ∈ ran 𝐹)))
1918ssopab2dv 5553 . . . 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 5684 . . . 4 ({𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} × ran 𝐹) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} ∧ 𝑦 ∈ ran 𝐹)}
2119, 20sseqtrrdi 4031 . . 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 7917 . . . . 5 dom 𝐹 ∈ V
2423abrexex 7966 . . . 4 {𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} ∈ V
2522rnex 7918 . . . 4 ran 𝐹 ∈ V
2624, 25xpex 7755 . . 3 ({𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} × ran 𝐹) ∈ V
27 ssexg 5323 . . 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 585 . 2 (𝐴 ∈ ℂ → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)} ∈ V)
29 breq 5150 . . . . . 6 (𝑧 = 𝐹 → ((𝑥𝑤)𝑧𝑦 ↔ (𝑥𝑤)𝐹𝑦))
3029anbi2d 629 . . . . 5 (𝑧 = 𝐹 → ((𝑥 ∈ ℂ ∧ (𝑥𝑤)𝑧𝑦) ↔ (𝑥 ∈ ℂ ∧ (𝑥𝑤)𝐹𝑦)))
3130opabbidv 5214 . . . 4 (𝑧 = 𝐹 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝑤)𝑧𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝑤)𝐹𝑦)})
32 oveq2 7428 . . . . . . 7 (𝑤 = 𝐴 → (𝑥𝑤) = (𝑥𝐴))
3332breq1d 5158 . . . . . 6 (𝑤 = 𝐴 → ((𝑥𝑤)𝐹𝑦 ↔ (𝑥𝐴)𝐹𝑦))
3433anbi2d 629 . . . . 5 (𝑤 = 𝐴 → ((𝑥 ∈ ℂ ∧ (𝑥𝑤)𝐹𝑦) ↔ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)))
3534opabbidv 5214 . . . 4 (𝑤 = 𝐴 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝑤)𝐹𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)})
36 df-shft 15046 . . . 4 shift = (𝑧 ∈ V, 𝑤 ∈ ℂ ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝑤)𝑧𝑦)})
3731, 35, 36ovmpog 7580 . . 3 ((𝐹 ∈ V ∧ 𝐴 ∈ ℂ ∧ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)} ∈ V) → (𝐹 shift 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)})
3822, 37mp3an1 1445 . 2 ((𝐴 ∈ ℂ ∧ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)} ∈ V) → (𝐹 shift 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)})
3928, 38mpdan 686 1 (𝐴 ∈ ℂ → (𝐹 shift 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  {cab 2705  wrex 3067  Vcvv 3471  wss 3947   class class class wbr 5148  {copab 5210   × cxp 5676  dom cdm 5678  ran crn 5679  (class class class)co 7420  cc 11136   + caddc 11141  cmin 11474   shift cshi 15045
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 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-po 5590  df-so 5591  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-er 8724  df-en 8964  df-dom 8965  df-sdom 8966  df-pnf 11280  df-mnf 11281  df-ltxr 11283  df-sub 11476  df-shft 15046
This theorem is referenced by:  shftdm  15050  shftfib  15051  shftfn  15052  2shfti  15059  shftidt2  15060
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