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Theorem shftfval 14964
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 7394 . . . . . . . . . 10 (𝑥𝐴) ∈ V
2 vex 3451 . . . . . . . . . 10 𝑦 ∈ V
31, 2breldm 5868 . . . . . . . . 9 ((𝑥𝐴)𝐹𝑦 → (𝑥𝐴) ∈ dom 𝐹)
4 npcan 11418 . . . . . . . . . . 11 ((𝑥 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((𝑥𝐴) + 𝐴) = 𝑥)
54eqcomd 2739 . . . . . . . . . 10 ((𝑥 ∈ ℂ ∧ 𝐴 ∈ ℂ) → 𝑥 = ((𝑥𝐴) + 𝐴))
65ancoms 460 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → 𝑥 = ((𝑥𝐴) + 𝐴))
7 oveq1 7368 . . . . . . . . . 10 (𝑤 = (𝑥𝐴) → (𝑤 + 𝐴) = ((𝑥𝐴) + 𝐴))
87rspceeqv 3599 . . . . . . . . 9 (((𝑥𝐴) ∈ dom 𝐹𝑥 = ((𝑥𝐴) + 𝐴)) → ∃𝑤 ∈ dom 𝐹 𝑥 = (𝑤 + 𝐴))
93, 6, 8syl2anr 598 . . . . . . . 8 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ (𝑥𝐴)𝐹𝑦) → ∃𝑤 ∈ dom 𝐹 𝑥 = (𝑤 + 𝐴))
10 vex 3451 . . . . . . . . 9 𝑥 ∈ V
11 eqeq1 2737 . . . . . . . . . 10 (𝑧 = 𝑥 → (𝑧 = (𝑤 + 𝐴) ↔ 𝑥 = (𝑤 + 𝐴)))
1211rexbidv 3172 . . . . . . . . 9 (𝑧 = 𝑥 → (∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴) ↔ ∃𝑤 ∈ dom 𝐹 𝑥 = (𝑤 + 𝐴)))
1310, 12elab 3634 . . . . . . . 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 5901 . . . . . . . 8 ((𝑥𝐴)𝐹𝑦𝑦 ∈ ran 𝐹)
1615adantl 483 . . . . . . 7 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ (𝑥𝐴)𝐹𝑦) → 𝑦 ∈ ran 𝐹)
1714, 16jca 513 . . . . . 6 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ (𝑥𝐴)𝐹𝑦) → (𝑥 ∈ {𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} ∧ 𝑦 ∈ ran 𝐹))
1817expl 459 . . . . 5 (𝐴 ∈ ℂ → ((𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦) → (𝑥 ∈ {𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} ∧ 𝑦 ∈ ran 𝐹)))
1918ssopab2dv 5512 . . . 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 5643 . . . 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 3999 . . 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 7852 . . . . 5 dom 𝐹 ∈ V
2423abrexex 7899 . . . 4 {𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} ∈ V
2522rnex 7853 . . . 4 ran 𝐹 ∈ V
2624, 25xpex 7691 . . 3 ({𝑧 ∣ ∃𝑤 ∈ dom 𝐹 𝑧 = (𝑤 + 𝐴)} × ran 𝐹) ∈ V
27 ssexg 5284 . . 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 587 . 2 (𝐴 ∈ ℂ → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)} ∈ V)
29 breq 5111 . . . . . 6 (𝑧 = 𝐹 → ((𝑥𝑤)𝑧𝑦 ↔ (𝑥𝑤)𝐹𝑦))
3029anbi2d 630 . . . . 5 (𝑧 = 𝐹 → ((𝑥 ∈ ℂ ∧ (𝑥𝑤)𝑧𝑦) ↔ (𝑥 ∈ ℂ ∧ (𝑥𝑤)𝐹𝑦)))
3130opabbidv 5175 . . . 4 (𝑧 = 𝐹 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝑤)𝑧𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝑤)𝐹𝑦)})
32 oveq2 7369 . . . . . . 7 (𝑤 = 𝐴 → (𝑥𝑤) = (𝑥𝐴))
3332breq1d 5119 . . . . . 6 (𝑤 = 𝐴 → ((𝑥𝑤)𝐹𝑦 ↔ (𝑥𝐴)𝐹𝑦))
3433anbi2d 630 . . . . 5 (𝑤 = 𝐴 → ((𝑥 ∈ ℂ ∧ (𝑥𝑤)𝐹𝑦) ↔ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)))
3534opabbidv 5175 . . . 4 (𝑤 = 𝐴 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝑤)𝐹𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)})
36 df-shft 14961 . . . 4 shift = (𝑧 ∈ V, 𝑤 ∈ ℂ ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝑤)𝑧𝑦)})
3731, 35, 36ovmpog 7518 . . 3 ((𝐹 ∈ V ∧ 𝐴 ∈ ℂ ∧ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)} ∈ V) → (𝐹 shift 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)})
3822, 37mp3an1 1449 . 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 397   = wceq 1542  wcel 2107  {cab 2710  wrex 3070  Vcvv 3447  wss 3914   class class class wbr 5109  {copab 5171   × cxp 5635  dom cdm 5637  ran crn 5638  (class class class)co 7361  cc 11057   + caddc 11062  cmin 11393   shift cshi 14960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-po 5549  df-so 5550  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-er 8654  df-en 8890  df-dom 8891  df-sdom 8892  df-pnf 11199  df-mnf 11200  df-ltxr 11202  df-sub 11395  df-shft 14961
This theorem is referenced by:  shftdm  14965  shftfib  14966  shftfn  14967  2shfti  14974  shftidt2  14975
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