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Theorem seqexw 13797
Description: Weak version of seqex 13783 that holds without ax-rep 5217. 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 13793 . . . 4 (𝑀 ∈ ℤ → seq𝑀( + , 𝐹) Fn (ℤ𝑀))
31, 2ax-mp 5 . . 3 seq𝑀( + , 𝐹) Fn (ℤ𝑀)
4 fnfun 6564 . . 3 (seq𝑀( + , 𝐹) Fn (ℤ𝑀) → Fun seq𝑀( + , 𝐹))
53, 4ax-mp 5 . 2 Fun seq𝑀( + , 𝐹)
63fndmi 6568 . . 3 dom seq𝑀( + , 𝐹) = (ℤ𝑀)
7 fvex 6817 . . 3 (ℤ𝑀) ∈ V
86, 7eqeltri 2832 . 2 dom seq𝑀( + , 𝐹) ∈ V
9 seqexw.1 . . . . 5 + ∈ V
109rnex 7795 . . . 4 ran + ∈ V
11 prex 5363 . . . 4 {∅, (𝐹𝑀)} ∈ V
1210, 11unex 7629 . . 3 (ran + ∪ {∅, (𝐹𝑀)}) ∈ V
13 fveq2 6804 . . . . . . 7 (𝑦 = 𝑀 → (seq𝑀( + , 𝐹)‘𝑦) = (seq𝑀( + , 𝐹)‘𝑀))
1413eleq1d 2820 . . . . . 6 (𝑦 = 𝑀 → ((seq𝑀( + , 𝐹)‘𝑦) ∈ (ran + ∪ {∅, (𝐹𝑀)}) ↔ (seq𝑀( + , 𝐹)‘𝑀) ∈ (ran + ∪ {∅, (𝐹𝑀)})))
15 fveq2 6804 . . . . . . 7 (𝑦 = 𝑧 → (seq𝑀( + , 𝐹)‘𝑦) = (seq𝑀( + , 𝐹)‘𝑧))
1615eleq1d 2820 . . . . . 6 (𝑦 = 𝑧 → ((seq𝑀( + , 𝐹)‘𝑦) ∈ (ran + ∪ {∅, (𝐹𝑀)}) ↔ (seq𝑀( + , 𝐹)‘𝑧) ∈ (ran + ∪ {∅, (𝐹𝑀)})))
17 fveq2 6804 . . . . . . 7 (𝑦 = (𝑧 + 1) → (seq𝑀( + , 𝐹)‘𝑦) = (seq𝑀( + , 𝐹)‘(𝑧 + 1)))
1817eleq1d 2820 . . . . . 6 (𝑦 = (𝑧 + 1) → ((seq𝑀( + , 𝐹)‘𝑦) ∈ (ran + ∪ {∅, (𝐹𝑀)}) ↔ (seq𝑀( + , 𝐹)‘(𝑧 + 1)) ∈ (ran + ∪ {∅, (𝐹𝑀)})))
19 fveq2 6804 . . . . . . 7 (𝑦 = 𝑥 → (seq𝑀( + , 𝐹)‘𝑦) = (seq𝑀( + , 𝐹)‘𝑥))
2019eleq1d 2820 . . . . . 6 (𝑦 = 𝑥 → ((seq𝑀( + , 𝐹)‘𝑦) ∈ (ran + ∪ {∅, (𝐹𝑀)}) ↔ (seq𝑀( + , 𝐹)‘𝑥) ∈ (ran + ∪ {∅, (𝐹𝑀)})))
21 seq1 13794 . . . . . . 7 (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹𝑀))
22 ssun2 4112 . . . . . . . 8 {∅, (𝐹𝑀)} ⊆ (ran + ∪ {∅, (𝐹𝑀)})
23 fvex 6817 . . . . . . . . 9 (𝐹𝑀) ∈ V
2423prid2 4702 . . . . . . . 8 (𝐹𝑀) ∈ {∅, (𝐹𝑀)}
2522, 24sselii 3922 . . . . . . 7 (𝐹𝑀) ∈ (ran + ∪ {∅, (𝐹𝑀)})
2621, 25eqeltrdi 2844 . . . . . 6 (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) ∈ (ran + ∪ {∅, (𝐹𝑀)}))
27 seqp1 13796 . . . . . . . . 9 (𝑧 ∈ (ℤ𝑀) → (seq𝑀( + , 𝐹)‘(𝑧 + 1)) = ((seq𝑀( + , 𝐹)‘𝑧) + (𝐹‘(𝑧 + 1))))
2827adantr 481 . . . . . . . 8 ((𝑧 ∈ (ℤ𝑀) ∧ (seq𝑀( + , 𝐹)‘𝑧) ∈ (ran + ∪ {∅, (𝐹𝑀)})) → (seq𝑀( + , 𝐹)‘(𝑧 + 1)) = ((seq𝑀( + , 𝐹)‘𝑧) + (𝐹‘(𝑧 + 1))))
29 df-ov 7311 . . . . . . . . . 10 ((seq𝑀( + , 𝐹)‘𝑧) + (𝐹‘(𝑧 + 1))) = ( + ‘⟨(seq𝑀( + , 𝐹)‘𝑧), (𝐹‘(𝑧 + 1))⟩)
30 snsspr1 4752 . . . . . . . . . . . 12 {∅} ⊆ {∅, (𝐹𝑀)}
31 unss2 4120 . . . . . . . . . . . 12 ({∅} ⊆ {∅, (𝐹𝑀)} → (ran + ∪ {∅}) ⊆ (ran + ∪ {∅, (𝐹𝑀)}))
3230, 31ax-mp 5 . . . . . . . . . . 11 (ran + ∪ {∅}) ⊆ (ran + ∪ {∅, (𝐹𝑀)})
33 fvrn0 6834 . . . . . . . . . . 11 ( + ‘⟨(seq𝑀( + , 𝐹)‘𝑧), (𝐹‘(𝑧 + 1))⟩) ∈ (ran + ∪ {∅})
3432, 33sselii 3922 . . . . . . . . . 10 ( + ‘⟨(seq𝑀( + , 𝐹)‘𝑧), (𝐹‘(𝑧 + 1))⟩) ∈ (ran + ∪ {∅, (𝐹𝑀)})
3529, 34eqeltri 2832 . . . . . . . . 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 2836 . . . . . . 7 ((𝑧 ∈ (ℤ𝑀) ∧ (seq𝑀( + , 𝐹)‘𝑧) ∈ (ran + ∪ {∅, (𝐹𝑀)})) → (seq𝑀( + , 𝐹)‘(𝑧 + 1)) ∈ (ran + ∪ {∅, (𝐹𝑀)}))
3837ex 413 . . . . . 6 (𝑧 ∈ (ℤ𝑀) → ((seq𝑀( + , 𝐹)‘𝑧) ∈ (ran + ∪ {∅, (𝐹𝑀)}) → (seq𝑀( + , 𝐹)‘(𝑧 + 1)) ∈ (ran + ∪ {∅, (𝐹𝑀)})))
3914, 16, 18, 20, 26, 38uzind4 12706 . . . . 5 (𝑥 ∈ (ℤ𝑀) → (seq𝑀( + , 𝐹)‘𝑥) ∈ (ran + ∪ {∅, (𝐹𝑀)}))
4039rgen 3062 . . . 4 𝑥 ∈ (ℤ𝑀)(seq𝑀( + , 𝐹)‘𝑥) ∈ (ran + ∪ {∅, (𝐹𝑀)})
41 fnfvrnss 7026 . . . 4 ((seq𝑀( + , 𝐹) Fn (ℤ𝑀) ∧ ∀𝑥 ∈ (ℤ𝑀)(seq𝑀( + , 𝐹)‘𝑥) ∈ (ran + ∪ {∅, (𝐹𝑀)})) → ran seq𝑀( + , 𝐹) ⊆ (ran + ∪ {∅, (𝐹𝑀)}))
423, 40, 41mp2an 689 . . 3 ran seq𝑀( + , 𝐹) ⊆ (ran + ∪ {∅, (𝐹𝑀)})
4312, 42ssexi 5254 . 2 ran seq𝑀( + , 𝐹) ∈ V
44 funexw 7830 . 2 ((Fun seq𝑀( + , 𝐹) ∧ dom seq𝑀( + , 𝐹) ∈ V ∧ ran seq𝑀( + , 𝐹) ∈ V) → seq𝑀( + , 𝐹) ∈ V)
455, 8, 43, 44mp3an 1460 1 seq𝑀( + , 𝐹) ∈ V
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1538  wcel 2103  wral 3060  Vcvv 3436  cun 3889  wss 3891  c0 4261  {csn 4564  {cpr 4566  cop 4570  dom cdm 5600  ran crn 5601  Fun wfun 6452   Fn wfn 6453  cfv 6458  (class class class)co 7308  1c1 10932   + caddc 10934  cz 12379  cuz 12642  seqcseq 13781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1968  ax-7 2008  ax-8 2105  ax-9 2113  ax-10 2134  ax-11 2151  ax-12 2168  ax-ext 2706  ax-sep 5231  ax-nul 5238  ax-pow 5296  ax-pr 5360  ax-un 7621  ax-cnex 10987  ax-resscn 10988  ax-1cn 10989  ax-icn 10990  ax-addcl 10991  ax-addrcl 10992  ax-mulcl 10993  ax-mulrcl 10994  ax-mulcom 10995  ax-addass 10996  ax-mulass 10997  ax-distr 10998  ax-i2m1 10999  ax-1ne0 11000  ax-1rid 11001  ax-rnegex 11002  ax-rrecex 11003  ax-cnre 11004  ax-pre-lttri 11005  ax-pre-lttrn 11006  ax-pre-ltadd 11007  ax-pre-mulgt0 11008
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1541  df-fal 1551  df-ex 1779  df-nf 1783  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2713  df-cleq 2727  df-clel 2813  df-nfc 2885  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3340  df-rab 3357  df-v 3438  df-sbc 3721  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-pss 3910  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4844  df-iun 4932  df-br 5081  df-opab 5143  df-mpt 5164  df-tr 5198  df-id 5500  df-eprel 5506  df-po 5514  df-so 5515  df-fr 5555  df-we 5557  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-pred 6217  df-ord 6284  df-on 6285  df-lim 6286  df-suc 6287  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-riota 7265  df-ov 7311  df-oprab 7312  df-mpo 7313  df-om 7749  df-2nd 7868  df-frecs 8132  df-wrecs 8163  df-recs 8237  df-rdg 8276  df-er 8534  df-en 8770  df-dom 8771  df-sdom 8772  df-pnf 11071  df-mnf 11072  df-xr 11073  df-ltxr 11074  df-le 11075  df-sub 11267  df-neg 11268  df-nn 12034  df-n0 12294  df-z 12380  df-uz 12643  df-seq 13782
This theorem is referenced by:  mulgfval  18761
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