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Theorem seqexw 14055
Description: Weak version of seqex 14041 that holds without ax-rep 5285. 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 14051 . . . 4 (𝑀 ∈ ℤ → seq𝑀( + , 𝐹) Fn (ℤ𝑀))
31, 2ax-mp 5 . . 3 seq𝑀( + , 𝐹) Fn (ℤ𝑀)
4 fnfun 6669 . . 3 (seq𝑀( + , 𝐹) Fn (ℤ𝑀) → Fun seq𝑀( + , 𝐹))
53, 4ax-mp 5 . 2 Fun seq𝑀( + , 𝐹)
63fndmi 6673 . . 3 dom seq𝑀( + , 𝐹) = (ℤ𝑀)
7 fvex 6920 . . 3 (ℤ𝑀) ∈ V
86, 7eqeltri 2835 . 2 dom seq𝑀( + , 𝐹) ∈ V
9 seqexw.1 . . . . 5 + ∈ V
109rnex 7933 . . . 4 ran + ∈ V
11 prex 5443 . . . 4 {∅, (𝐹𝑀)} ∈ V
1210, 11unex 7763 . . 3 (ran + ∪ {∅, (𝐹𝑀)}) ∈ V
13 fveq2 6907 . . . . . . 7 (𝑦 = 𝑀 → (seq𝑀( + , 𝐹)‘𝑦) = (seq𝑀( + , 𝐹)‘𝑀))
1413eleq1d 2824 . . . . . 6 (𝑦 = 𝑀 → ((seq𝑀( + , 𝐹)‘𝑦) ∈ (ran + ∪ {∅, (𝐹𝑀)}) ↔ (seq𝑀( + , 𝐹)‘𝑀) ∈ (ran + ∪ {∅, (𝐹𝑀)})))
15 fveq2 6907 . . . . . . 7 (𝑦 = 𝑧 → (seq𝑀( + , 𝐹)‘𝑦) = (seq𝑀( + , 𝐹)‘𝑧))
1615eleq1d 2824 . . . . . 6 (𝑦 = 𝑧 → ((seq𝑀( + , 𝐹)‘𝑦) ∈ (ran + ∪ {∅, (𝐹𝑀)}) ↔ (seq𝑀( + , 𝐹)‘𝑧) ∈ (ran + ∪ {∅, (𝐹𝑀)})))
17 fveq2 6907 . . . . . . 7 (𝑦 = (𝑧 + 1) → (seq𝑀( + , 𝐹)‘𝑦) = (seq𝑀( + , 𝐹)‘(𝑧 + 1)))
1817eleq1d 2824 . . . . . 6 (𝑦 = (𝑧 + 1) → ((seq𝑀( + , 𝐹)‘𝑦) ∈ (ran + ∪ {∅, (𝐹𝑀)}) ↔ (seq𝑀( + , 𝐹)‘(𝑧 + 1)) ∈ (ran + ∪ {∅, (𝐹𝑀)})))
19 fveq2 6907 . . . . . . 7 (𝑦 = 𝑥 → (seq𝑀( + , 𝐹)‘𝑦) = (seq𝑀( + , 𝐹)‘𝑥))
2019eleq1d 2824 . . . . . 6 (𝑦 = 𝑥 → ((seq𝑀( + , 𝐹)‘𝑦) ∈ (ran + ∪ {∅, (𝐹𝑀)}) ↔ (seq𝑀( + , 𝐹)‘𝑥) ∈ (ran + ∪ {∅, (𝐹𝑀)})))
21 seq1 14052 . . . . . . 7 (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹𝑀))
22 ssun2 4189 . . . . . . . 8 {∅, (𝐹𝑀)} ⊆ (ran + ∪ {∅, (𝐹𝑀)})
23 fvex 6920 . . . . . . . . 9 (𝐹𝑀) ∈ V
2423prid2 4768 . . . . . . . 8 (𝐹𝑀) ∈ {∅, (𝐹𝑀)}
2522, 24sselii 3992 . . . . . . 7 (𝐹𝑀) ∈ (ran + ∪ {∅, (𝐹𝑀)})
2621, 25eqeltrdi 2847 . . . . . 6 (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) ∈ (ran + ∪ {∅, (𝐹𝑀)}))
27 seqp1 14054 . . . . . . . . 9 (𝑧 ∈ (ℤ𝑀) → (seq𝑀( + , 𝐹)‘(𝑧 + 1)) = ((seq𝑀( + , 𝐹)‘𝑧) + (𝐹‘(𝑧 + 1))))
2827adantr 480 . . . . . . . 8 ((𝑧 ∈ (ℤ𝑀) ∧ (seq𝑀( + , 𝐹)‘𝑧) ∈ (ran + ∪ {∅, (𝐹𝑀)})) → (seq𝑀( + , 𝐹)‘(𝑧 + 1)) = ((seq𝑀( + , 𝐹)‘𝑧) + (𝐹‘(𝑧 + 1))))
29 df-ov 7434 . . . . . . . . . 10 ((seq𝑀( + , 𝐹)‘𝑧) + (𝐹‘(𝑧 + 1))) = ( + ‘⟨(seq𝑀( + , 𝐹)‘𝑧), (𝐹‘(𝑧 + 1))⟩)
30 snsspr1 4819 . . . . . . . . . . . 12 {∅} ⊆ {∅, (𝐹𝑀)}
31 unss2 4197 . . . . . . . . . . . 12 ({∅} ⊆ {∅, (𝐹𝑀)} → (ran + ∪ {∅}) ⊆ (ran + ∪ {∅, (𝐹𝑀)}))
3230, 31ax-mp 5 . . . . . . . . . . 11 (ran + ∪ {∅}) ⊆ (ran + ∪ {∅, (𝐹𝑀)})
33 fvrn0 6937 . . . . . . . . . . 11 ( + ‘⟨(seq𝑀( + , 𝐹)‘𝑧), (𝐹‘(𝑧 + 1))⟩) ∈ (ran + ∪ {∅})
3432, 33sselii 3992 . . . . . . . . . 10 ( + ‘⟨(seq𝑀( + , 𝐹)‘𝑧), (𝐹‘(𝑧 + 1))⟩) ∈ (ran + ∪ {∅, (𝐹𝑀)})
3529, 34eqeltri 2835 . . . . . . . . 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 2839 . . . . . . 7 ((𝑧 ∈ (ℤ𝑀) ∧ (seq𝑀( + , 𝐹)‘𝑧) ∈ (ran + ∪ {∅, (𝐹𝑀)})) → (seq𝑀( + , 𝐹)‘(𝑧 + 1)) ∈ (ran + ∪ {∅, (𝐹𝑀)}))
3837ex 412 . . . . . 6 (𝑧 ∈ (ℤ𝑀) → ((seq𝑀( + , 𝐹)‘𝑧) ∈ (ran + ∪ {∅, (𝐹𝑀)}) → (seq𝑀( + , 𝐹)‘(𝑧 + 1)) ∈ (ran + ∪ {∅, (𝐹𝑀)})))
3914, 16, 18, 20, 26, 38uzind4 12946 . . . . 5 (𝑥 ∈ (ℤ𝑀) → (seq𝑀( + , 𝐹)‘𝑥) ∈ (ran + ∪ {∅, (𝐹𝑀)}))
4039rgen 3061 . . . 4 𝑥 ∈ (ℤ𝑀)(seq𝑀( + , 𝐹)‘𝑥) ∈ (ran + ∪ {∅, (𝐹𝑀)})
41 fnfvrnss 7141 . . . 4 ((seq𝑀( + , 𝐹) Fn (ℤ𝑀) ∧ ∀𝑥 ∈ (ℤ𝑀)(seq𝑀( + , 𝐹)‘𝑥) ∈ (ran + ∪ {∅, (𝐹𝑀)})) → ran seq𝑀( + , 𝐹) ⊆ (ran + ∪ {∅, (𝐹𝑀)}))
423, 40, 41mp2an 692 . . 3 ran seq𝑀( + , 𝐹) ⊆ (ran + ∪ {∅, (𝐹𝑀)})
4312, 42ssexi 5328 . 2 ran seq𝑀( + , 𝐹) ∈ V
44 funexw 7975 . 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 395   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  cun 3961  wss 3963  c0 4339  {csn 4631  {cpr 4633  cop 4637  dom cdm 5689  ran crn 5690  Fun wfun 6557   Fn wfn 6558  cfv 6563  (class class class)co 7431  1c1 11154   + caddc 11156  cz 12611  cuz 12876  seqcseq 14039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-seq 14040
This theorem is referenced by:  mulgfval  19100
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