Theorem shftfn 11350

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 𝐵 ∧ 𝐴 ∈ ℂ) → (𝐹 shift 𝐴) Fn {𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ 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.

Step	Hyp	Ref	Expression
1		relopab 4848	. . . . 5 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)}
2	1	a1i 9	. . . 4 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐴 ∈ ℂ) → Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)})
3		fnfun 5418	. . . . . 6 ⊢ (𝐹 Fn 𝐵 → Fun 𝐹)
4	3	adantr 276	. . . . 5 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐴 ∈ ℂ) → Fun 𝐹)
5		funmo 5333	. . . . . . 7 ⊢ (Fun 𝐹 → ∃*𝑤(𝑧 − 𝐴)𝐹𝑤)
6		vex 2802	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
7		vex 2802	. . . . . . . . . 10 ⊢ 𝑤 ∈ V
8		eleq1 2292	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ ℂ ↔ 𝑧 ∈ ℂ))
9		oveq1 6014	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑥 − 𝐴) = (𝑧 − 𝐴))
10	9	breq1d 4093	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → ((𝑥 − 𝐴)𝐹𝑦 ↔ (𝑧 − 𝐴)𝐹𝑦))
11	8, 10	anbi12d 473	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦) ↔ (𝑧 ∈ ℂ ∧ (𝑧 − 𝐴)𝐹𝑦)))
12		breq2 4087	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑤 → ((𝑧 − 𝐴)𝐹𝑦 ↔ (𝑧 − 𝐴)𝐹𝑤))
13	12	anbi2d 464	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → ((𝑧 ∈ ℂ ∧ (𝑧 − 𝐴)𝐹𝑦) ↔ (𝑧 ∈ ℂ ∧ (𝑧 − 𝐴)𝐹𝑤)))
14		eqid 2229	. . . . . . . . . 10 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)}
15	6, 7, 11, 13, 14	brab 4361	. . . . . . . . 9 ⊢ (𝑧{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)}𝑤 ↔ (𝑧 ∈ ℂ ∧ (𝑧 − 𝐴)𝐹𝑤))
16	15	simprbi 275	. . . . . . . 8 ⊢ (𝑧{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)}𝑤 → (𝑧 − 𝐴)𝐹𝑤)
17	16	moimi 2143	. . . . . . 7 ⊢ (∃*𝑤(𝑧 − 𝐴)𝐹𝑤 → ∃*𝑤 𝑧{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)}𝑤)
18	5, 17	syl 14	. . . . . 6 ⊢ (Fun 𝐹 → ∃*𝑤 𝑧{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)}𝑤)
19	18	alrimiv 1920	. . . . 5 ⊢ (Fun 𝐹 → ∀𝑧∃*𝑤 𝑧{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)}𝑤)
20	4, 19	syl 14	. . . 4 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐴 ∈ ℂ) → ∀𝑧∃*𝑤 𝑧{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)}𝑤)
21		dffun6 5332	. . . 4 ⊢ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)} ↔ (Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)} ∧ ∀𝑧∃*𝑤 𝑧{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)}𝑤))
22	2, 20, 21	sylanbrc 417	. . 3 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐴 ∈ ℂ) → Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)})
23		shftfval.1	. . . . . 6 ⊢ 𝐹 ∈ V
24	23	shftfval 11347	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝐹 shift 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)})
25	24	adantl 277	. . . 4 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐴 ∈ ℂ) → (𝐹 shift 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)})
26	25	funeqd 5340	. . 3 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐴 ∈ ℂ) → (Fun (𝐹 shift 𝐴) ↔ Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)}))
27	22, 26	mpbird 167	. 2 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐴 ∈ ℂ) → Fun (𝐹 shift 𝐴))
28	23	shftdm 11348	. . 3 ⊢ (𝐴 ∈ ℂ → dom (𝐹 shift 𝐴) = {𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ dom 𝐹})
29		fndm 5420	. . . . 5 ⊢ (𝐹 Fn 𝐵 → dom 𝐹 = 𝐵)
30	29	eleq2d 2299	. . . 4 ⊢ (𝐹 Fn 𝐵 → ((𝑥 − 𝐴) ∈ dom 𝐹 ↔ (𝑥 − 𝐴) ∈ 𝐵))
31	30	rabbidv 2788	. . 3 ⊢ (𝐹 Fn 𝐵 → {𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ dom 𝐹} = {𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ 𝐵})
32	28, 31	sylan9eqr 2284	. 2 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐴 ∈ ℂ) → dom (𝐹 shift 𝐴) = {𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ 𝐵})
33		df-fn 5321	. 2 ⊢ ((𝐹 shift 𝐴) Fn {𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ 𝐵} ↔ (Fun (𝐹 shift 𝐴) ∧ dom (𝐹 shift 𝐴) = {𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ 𝐵}))
34	27, 32, 33	sylanbrc 417	1 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐴 ∈ ℂ) → (𝐹 shift 𝐴) Fn {𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ 𝐵})

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1393   = wceq 1395  ∃*wmo 2078   ∈ wcel 2200  {crab 2512  Vcvv 2799   class class class wbr 4083  {copab 4144  dom cdm 4719  Rel wrel 4724  Fun wfun 5312   Fn wfn 5313  (class class class)co 6007  ℂcc 8008   − cmin 8328   shift cshi 11340

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-resscn 8102  ax-1cn 8103  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-addcom 8110  ax-addass 8112  ax-distr 8114  ax-i2m1 8115  ax-0id 8118  ax-rnegex 8119  ax-cnre 8121

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-sub 8330  df-shft 11341

This theorem is referenced by:  shftf  11356  seq3shft  11364