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Theorem seqexw 13400
 Description: Weak version of seqex 13386 that holds without ax-rep 5158. 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.
StepHypRef Expression
1 seqexw.2 . . . 4 𝑀 ∈ ℤ
2 seqfn 13396 . . . 4 (𝑀 ∈ ℤ → seq𝑀( + , 𝐹) Fn (ℤ𝑀))
31, 2ax-mp 5 . . 3 seq𝑀( + , 𝐹) Fn (ℤ𝑀)
4 fnfun 6431 . . 3 (seq𝑀( + , 𝐹) Fn (ℤ𝑀) → Fun seq𝑀( + , 𝐹))
53, 4ax-mp 5 . 2 Fun seq𝑀( + , 𝐹)
63fndmi 6434 . . 3 dom seq𝑀( + , 𝐹) = (ℤ𝑀)
7 fvex 6668 . . 3 (ℤ𝑀) ∈ V
86, 7eqeltri 2886 . 2 dom seq𝑀( + , 𝐹) ∈ V
9 seqexw.1 . . . . 5 + ∈ V
109rnex 7612 . . . 4 ran + ∈ V
11 prex 5302 . . . 4 {∅, (𝐹𝑀)} ∈ V
1210, 11unex 7462 . . 3 (ran + ∪ {∅, (𝐹𝑀)}) ∈ V
13 fveq2 6655 . . . . . . 7 (𝑦 = 𝑀 → (seq𝑀( + , 𝐹)‘𝑦) = (seq𝑀( + , 𝐹)‘𝑀))
1413eleq1d 2874 . . . . . 6 (𝑦 = 𝑀 → ((seq𝑀( + , 𝐹)‘𝑦) ∈ (ran + ∪ {∅, (𝐹𝑀)}) ↔ (seq𝑀( + , 𝐹)‘𝑀) ∈ (ran + ∪ {∅, (𝐹𝑀)})))
15 fveq2 6655 . . . . . . 7 (𝑦 = 𝑧 → (seq𝑀( + , 𝐹)‘𝑦) = (seq𝑀( + , 𝐹)‘𝑧))
1615eleq1d 2874 . . . . . 6 (𝑦 = 𝑧 → ((seq𝑀( + , 𝐹)‘𝑦) ∈ (ran + ∪ {∅, (𝐹𝑀)}) ↔ (seq𝑀( + , 𝐹)‘𝑧) ∈ (ran + ∪ {∅, (𝐹𝑀)})))
17 fveq2 6655 . . . . . . 7 (𝑦 = (𝑧 + 1) → (seq𝑀( + , 𝐹)‘𝑦) = (seq𝑀( + , 𝐹)‘(𝑧 + 1)))
1817eleq1d 2874 . . . . . 6 (𝑦 = (𝑧 + 1) → ((seq𝑀( + , 𝐹)‘𝑦) ∈ (ran + ∪ {∅, (𝐹𝑀)}) ↔ (seq𝑀( + , 𝐹)‘(𝑧 + 1)) ∈ (ran + ∪ {∅, (𝐹𝑀)})))
19 fveq2 6655 . . . . . . 7 (𝑦 = 𝑥 → (seq𝑀( + , 𝐹)‘𝑦) = (seq𝑀( + , 𝐹)‘𝑥))
2019eleq1d 2874 . . . . . 6 (𝑦 = 𝑥 → ((seq𝑀( + , 𝐹)‘𝑦) ∈ (ran + ∪ {∅, (𝐹𝑀)}) ↔ (seq𝑀( + , 𝐹)‘𝑥) ∈ (ran + ∪ {∅, (𝐹𝑀)})))
21 seq1 13397 . . . . . . 7 (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹𝑀))
22 ssun2 4103 . . . . . . . 8 {∅, (𝐹𝑀)} ⊆ (ran + ∪ {∅, (𝐹𝑀)})
23 fvex 6668 . . . . . . . . 9 (𝐹𝑀) ∈ V
2423prid2 4662 . . . . . . . 8 (𝐹𝑀) ∈ {∅, (𝐹𝑀)}
2522, 24sselii 3914 . . . . . . 7 (𝐹𝑀) ∈ (ran + ∪ {∅, (𝐹𝑀)})
2621, 25eqeltrdi 2898 . . . . . 6 (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) ∈ (ran + ∪ {∅, (𝐹𝑀)}))
27 seqp1 13399 . . . . . . . . 9 (𝑧 ∈ (ℤ𝑀) → (seq𝑀( + , 𝐹)‘(𝑧 + 1)) = ((seq𝑀( + , 𝐹)‘𝑧) + (𝐹‘(𝑧 + 1))))
2827adantr 484 . . . . . . . 8 ((𝑧 ∈ (ℤ𝑀) ∧ (seq𝑀( + , 𝐹)‘𝑧) ∈ (ran + ∪ {∅, (𝐹𝑀)})) → (seq𝑀( + , 𝐹)‘(𝑧 + 1)) = ((seq𝑀( + , 𝐹)‘𝑧) + (𝐹‘(𝑧 + 1))))
29 df-ov 7148 . . . . . . . . . 10 ((seq𝑀( + , 𝐹)‘𝑧) + (𝐹‘(𝑧 + 1))) = ( + ‘⟨(seq𝑀( + , 𝐹)‘𝑧), (𝐹‘(𝑧 + 1))⟩)
30 snsspr1 4710 . . . . . . . . . . . 12 {∅} ⊆ {∅, (𝐹𝑀)}
31 unss2 4111 . . . . . . . . . . . 12 ({∅} ⊆ {∅, (𝐹𝑀)} → (ran + ∪ {∅}) ⊆ (ran + ∪ {∅, (𝐹𝑀)}))
3230, 31ax-mp 5 . . . . . . . . . . 11 (ran + ∪ {∅}) ⊆ (ran + ∪ {∅, (𝐹𝑀)})
33 fvrn0 6683 . . . . . . . . . . 11 ( + ‘⟨(seq𝑀( + , 𝐹)‘𝑧), (𝐹‘(𝑧 + 1))⟩) ∈ (ran + ∪ {∅})
3432, 33sselii 3914 . . . . . . . . . 10 ( + ‘⟨(seq𝑀( + , 𝐹)‘𝑧), (𝐹‘(𝑧 + 1))⟩) ∈ (ran + ∪ {∅, (𝐹𝑀)})
3529, 34eqeltri 2886 . . . . . . . . 9 ((seq𝑀( + , 𝐹)‘𝑧) + (𝐹‘(𝑧 + 1))) ∈ (ran + ∪ {∅, (𝐹𝑀)})
3635a1i 11 . . . . . . . 8 ((𝑧 ∈ (ℤ𝑀) ∧ (seq𝑀( + , 𝐹)‘𝑧) ∈ (ran + ∪ {∅, (𝐹𝑀)})) → ((seq𝑀( + , 𝐹)‘𝑧) + (𝐹‘(𝑧 + 1))) ∈ (ran + ∪ {∅, (𝐹𝑀)}))
3728, 36eqeltrd 2890 . . . . . . 7 ((𝑧 ∈ (ℤ𝑀) ∧ (seq𝑀( + , 𝐹)‘𝑧) ∈ (ran + ∪ {∅, (𝐹𝑀)})) → (seq𝑀( + , 𝐹)‘(𝑧 + 1)) ∈ (ran + ∪ {∅, (𝐹𝑀)}))
3837ex 416 . . . . . 6 (𝑧 ∈ (ℤ𝑀) → ((seq𝑀( + , 𝐹)‘𝑧) ∈ (ran + ∪ {∅, (𝐹𝑀)}) → (seq𝑀( + , 𝐹)‘(𝑧 + 1)) ∈ (ran + ∪ {∅, (𝐹𝑀)})))
3914, 16, 18, 20, 26, 38uzind4 12314 . . . . 5 (𝑥 ∈ (ℤ𝑀) → (seq𝑀( + , 𝐹)‘𝑥) ∈ (ran + ∪ {∅, (𝐹𝑀)}))
4039rgen 3116 . . . 4 𝑥 ∈ (ℤ𝑀)(seq𝑀( + , 𝐹)‘𝑥) ∈ (ran + ∪ {∅, (𝐹𝑀)})
41 fnfvrnss 6871 . . . 4 ((seq𝑀( + , 𝐹) Fn (ℤ𝑀) ∧ ∀𝑥 ∈ (ℤ𝑀)(seq𝑀( + , 𝐹)‘𝑥) ∈ (ran + ∪ {∅, (𝐹𝑀)})) → ran seq𝑀( + , 𝐹) ⊆ (ran + ∪ {∅, (𝐹𝑀)}))
423, 40, 41mp2an 691 . . 3 ran seq𝑀( + , 𝐹) ⊆ (ran + ∪ {∅, (𝐹𝑀)})
4312, 42ssexi 5194 . 2 ran seq𝑀( + , 𝐹) ∈ V
44 funexw 7648 . 2 ((Fun seq𝑀( + , 𝐹) ∧ dom seq𝑀( + , 𝐹) ∈ V ∧ ran seq𝑀( + , 𝐹) ∈ V) → seq𝑀( + , 𝐹) ∈ V)
455, 8, 43, 44mp3an 1458 1 seq𝑀( + , 𝐹) ∈ V
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  Vcvv 3442   ∪ cun 3881   ⊆ wss 3883  ∅c0 4246  {csn 4528  {cpr 4530  ⟨cop 4534  dom cdm 5523  ran crn 5524  Fun wfun 6326   Fn wfn 6327  ‘cfv 6332  (class class class)co 7145  1c1 10545   + caddc 10547  ℤcz 11989  ℤ≥cuz 12251  seqcseq 13384 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454  ax-cnex 10600  ax-resscn 10601  ax-1cn 10602  ax-icn 10603  ax-addcl 10604  ax-addrcl 10605  ax-mulcl 10606  ax-mulrcl 10607  ax-mulcom 10608  ax-addass 10609  ax-mulass 10610  ax-distr 10611  ax-i2m1 10612  ax-1ne0 10613  ax-1rid 10614  ax-rnegex 10615  ax-rrecex 10616  ax-cnre 10617  ax-pre-lttri 10618  ax-pre-lttrn 10619  ax-pre-ltadd 10620  ax-pre-mulgt0 10621 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4805  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-tr 5141  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7574  df-2nd 7685  df-wrecs 7948  df-recs 8009  df-rdg 8047  df-er 8290  df-en 8511  df-dom 8512  df-sdom 8513  df-pnf 10684  df-mnf 10685  df-xr 10686  df-ltxr 10687  df-le 10688  df-sub 10879  df-neg 10880  df-nn 11644  df-n0 11904  df-z 11990  df-uz 12252  df-seq 13385 This theorem is referenced by:  mulgfval  18239
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