Theorem seqexw 13665

Description: Weak version of seqex 13651 that holds without ax-rep 5205. A sequence builder exists when its binary operation input exists and its starting index is an integer. (Contributed by Rohan Ridenour, 14-Aug-2023.)

Hypotheses

Ref	Expression
seqexw.1	⊢ + ∈ V
seqexw.2	⊢ 𝑀 ∈ ℤ

Assertion

Ref	Expression
seqexw	⊢ seq𝑀( + , 𝐹) ∈ V

Proof of Theorem seqexw

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		seqexw.2	. . . 4 ⊢ 𝑀 ∈ ℤ
2		seqfn 13661	. . . 4 ⊢ (𝑀 ∈ ℤ → seq𝑀( + , 𝐹) Fn (ℤ≥‘𝑀))
3	1, 2	ax-mp 5	. . 3 ⊢ seq𝑀( + , 𝐹) Fn (ℤ≥‘𝑀)
4		fnfun 6517	. . 3 ⊢ (seq𝑀( + , 𝐹) Fn (ℤ≥‘𝑀) → Fun seq𝑀( + , 𝐹))
5	3, 4	ax-mp 5	. 2 ⊢ Fun seq𝑀( + , 𝐹)
6	3	fndmi 6521	. . 3 ⊢ dom seq𝑀( + , 𝐹) = (ℤ≥‘𝑀)
7		fvex 6769	. . 3 ⊢ (ℤ≥‘𝑀) ∈ V
8	6, 7	eqeltri 2835	. 2 ⊢ dom seq𝑀( + , 𝐹) ∈ V
9		seqexw.1	. . . . 5 ⊢ + ∈ V
10	9	rnex 7733	. . . 4 ⊢ ran + ∈ V
11		prex 5350	. . . 4 ⊢ {∅, (𝐹‘𝑀)} ∈ V
12	10, 11	unex 7574	. . 3 ⊢ (ran + ∪ {∅, (𝐹‘𝑀)}) ∈ V
13		fveq2 6756	. . . . . . 7 ⊢ (𝑦 = 𝑀 → (seq𝑀( + , 𝐹)‘𝑦) = (seq𝑀( + , 𝐹)‘𝑀))
14	13	eleq1d 2823	. . . . . 6 ⊢ (𝑦 = 𝑀 → ((seq𝑀( + , 𝐹)‘𝑦) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}) ↔ (seq𝑀( + , 𝐹)‘𝑀) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})))
15		fveq2 6756	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (seq𝑀( + , 𝐹)‘𝑦) = (seq𝑀( + , 𝐹)‘𝑧))
16	15	eleq1d 2823	. . . . . 6 ⊢ (𝑦 = 𝑧 → ((seq𝑀( + , 𝐹)‘𝑦) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}) ↔ (seq𝑀( + , 𝐹)‘𝑧) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})))
17		fveq2 6756	. . . . . . 7 ⊢ (𝑦 = (𝑧 + 1) → (seq𝑀( + , 𝐹)‘𝑦) = (seq𝑀( + , 𝐹)‘(𝑧 + 1)))
18	17	eleq1d 2823	. . . . . 6 ⊢ (𝑦 = (𝑧 + 1) → ((seq𝑀( + , 𝐹)‘𝑦) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}) ↔ (seq𝑀( + , 𝐹)‘(𝑧 + 1)) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})))
19		fveq2 6756	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (seq𝑀( + , 𝐹)‘𝑦) = (seq𝑀( + , 𝐹)‘𝑥))
20	19	eleq1d 2823	. . . . . 6 ⊢ (𝑦 = 𝑥 → ((seq𝑀( + , 𝐹)‘𝑦) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}) ↔ (seq𝑀( + , 𝐹)‘𝑥) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})))
21		seq1 13662	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀))
22		ssun2 4103	. . . . . . . 8 ⊢ {∅, (𝐹‘𝑀)} ⊆ (ran + ∪ {∅, (𝐹‘𝑀)})
23		fvex 6769	. . . . . . . . 9 ⊢ (𝐹‘𝑀) ∈ V
24	23	prid2 4696	. . . . . . . 8 ⊢ (𝐹‘𝑀) ∈ {∅, (𝐹‘𝑀)}
25	22, 24	sselii 3914	. . . . . . 7 ⊢ (𝐹‘𝑀) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})
26	21, 25	eqeltrdi 2847	. . . . . 6 ⊢ (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}))
27		seqp1 13664	. . . . . . . . 9 ⊢ (𝑧 ∈ (ℤ≥‘𝑀) → (seq𝑀( + , 𝐹)‘(𝑧 + 1)) = ((seq𝑀( + , 𝐹)‘𝑧) + (𝐹‘(𝑧 + 1))))
28	27	adantr 480	. . . . . . . 8 ⊢ ((𝑧 ∈ (ℤ≥‘𝑀) ∧ (seq𝑀( + , 𝐹)‘𝑧) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})) → (seq𝑀( + , 𝐹)‘(𝑧 + 1)) = ((seq𝑀( + , 𝐹)‘𝑧) + (𝐹‘(𝑧 + 1))))
29		df-ov 7258	. . . . . . . . . 10 ⊢ ((seq𝑀( + , 𝐹)‘𝑧) + (𝐹‘(𝑧 + 1))) = ( + ‘⟨(seq𝑀( + , 𝐹)‘𝑧), (𝐹‘(𝑧 + 1))⟩)
30		snsspr1 4744	. . . . . . . . . . . 12 ⊢ {∅} ⊆ {∅, (𝐹‘𝑀)}
31		unss2 4111	. . . . . . . . . . . 12 ⊢ ({∅} ⊆ {∅, (𝐹‘𝑀)} → (ran + ∪ {∅}) ⊆ (ran + ∪ {∅, (𝐹‘𝑀)}))
32	30, 31	ax-mp 5	. . . . . . . . . . 11 ⊢ (ran + ∪ {∅}) ⊆ (ran + ∪ {∅, (𝐹‘𝑀)})
33		fvrn0 6784	. . . . . . . . . . 11 ⊢ ( + ‘⟨(seq𝑀( + , 𝐹)‘𝑧), (𝐹‘(𝑧 + 1))⟩) ∈ (ran + ∪ {∅})
34	32, 33	sselii 3914	. . . . . . . . . 10 ⊢ ( + ‘⟨(seq𝑀( + , 𝐹)‘𝑧), (𝐹‘(𝑧 + 1))⟩) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})
35	29, 34	eqeltri 2835	. . . . . . . . 9 ⊢ ((seq𝑀( + , 𝐹)‘𝑧) + (𝐹‘(𝑧 + 1))) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})
36	35	a1i 11	. . . . . . . 8 ⊢ ((𝑧 ∈ (ℤ≥‘𝑀) ∧ (seq𝑀( + , 𝐹)‘𝑧) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})) → ((seq𝑀( + , 𝐹)‘𝑧) + (𝐹‘(𝑧 + 1))) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}))
37	28, 36	eqeltrd 2839	. . . . . . 7 ⊢ ((𝑧 ∈ (ℤ≥‘𝑀) ∧ (seq𝑀( + , 𝐹)‘𝑧) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})) → (seq𝑀( + , 𝐹)‘(𝑧 + 1)) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}))
38	37	ex 412	. . . . . 6 ⊢ (𝑧 ∈ (ℤ≥‘𝑀) → ((seq𝑀( + , 𝐹)‘𝑧) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}) → (seq𝑀( + , 𝐹)‘(𝑧 + 1)) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})))
39	14, 16, 18, 20, 26, 38	uzind4 12575	. . . . 5 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → (seq𝑀( + , 𝐹)‘𝑥) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}))
40	39	rgen 3073	. . . 4 ⊢ ∀𝑥 ∈ (ℤ≥‘𝑀)(seq𝑀( + , 𝐹)‘𝑥) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})
41		fnfvrnss 6976	. . . 4 ⊢ ((seq𝑀( + , 𝐹) Fn (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (ℤ≥‘𝑀)(seq𝑀( + , 𝐹)‘𝑥) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})) → ran seq𝑀( + , 𝐹) ⊆ (ran + ∪ {∅, (𝐹‘𝑀)}))
42	3, 40, 41	mp2an 688	. . 3 ⊢ ran seq𝑀( + , 𝐹) ⊆ (ran + ∪ {∅, (𝐹‘𝑀)})
43	12, 42	ssexi 5241	. 2 ⊢ ran seq𝑀( + , 𝐹) ∈ V
44		funexw 7768	. 2 ⊢ ((Fun seq𝑀( + , 𝐹) ∧ dom seq𝑀( + , 𝐹) ∈ V ∧ ran seq𝑀( + , 𝐹) ∈ V) → seq𝑀( + , 𝐹) ∈ V)
45	5, 8, 43, 44	mp3an 1459	1 ⊢ seq𝑀( + , 𝐹) ∈ V

Syntax hints:   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  Vcvv 3422   ∪ cun 3881   ⊆ wss 3883  ∅c0 4253  {csn 4558  {cpr 4560  ⟨cop 4564  dom cdm 5580  ran crn 5581  Fun wfun 6412   Fn wfn 6413  ‘cfv 6418  (class class class)co 7255  1c1 10803   + caddc 10805  ℤcz 12249  ℤ_≥cuz 12511  seqcseq 13649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-n0 12164  df-z 12250  df-uz 12512  df-seq 13650

This theorem is referenced by:  mulgfval  18617