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Theorem seqexw 13922
Description: Weak version of seqex 13908 that holds without ax-rep 5242. 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 13918 . . . 4 (𝑀 ∈ ℤ → seq𝑀( + , 𝐹) Fn (ℤ𝑀))
31, 2ax-mp 5 . . 3 seq𝑀( + , 𝐹) Fn (ℤ𝑀)
4 fnfun 6602 . . 3 (seq𝑀( + , 𝐹) Fn (ℤ𝑀) → Fun seq𝑀( + , 𝐹))
53, 4ax-mp 5 . 2 Fun seq𝑀( + , 𝐹)
63fndmi 6606 . . 3 dom seq𝑀( + , 𝐹) = (ℤ𝑀)
7 fvex 6855 . . 3 (ℤ𝑀) ∈ V
86, 7eqeltri 2834 . 2 dom seq𝑀( + , 𝐹) ∈ V
9 seqexw.1 . . . . 5 + ∈ V
109rnex 7849 . . . 4 ran + ∈ V
11 prex 5389 . . . 4 {∅, (𝐹𝑀)} ∈ V
1210, 11unex 7680 . . 3 (ran + ∪ {∅, (𝐹𝑀)}) ∈ V
13 fveq2 6842 . . . . . . 7 (𝑦 = 𝑀 → (seq𝑀( + , 𝐹)‘𝑦) = (seq𝑀( + , 𝐹)‘𝑀))
1413eleq1d 2822 . . . . . 6 (𝑦 = 𝑀 → ((seq𝑀( + , 𝐹)‘𝑦) ∈ (ran + ∪ {∅, (𝐹𝑀)}) ↔ (seq𝑀( + , 𝐹)‘𝑀) ∈ (ran + ∪ {∅, (𝐹𝑀)})))
15 fveq2 6842 . . . . . . 7 (𝑦 = 𝑧 → (seq𝑀( + , 𝐹)‘𝑦) = (seq𝑀( + , 𝐹)‘𝑧))
1615eleq1d 2822 . . . . . 6 (𝑦 = 𝑧 → ((seq𝑀( + , 𝐹)‘𝑦) ∈ (ran + ∪ {∅, (𝐹𝑀)}) ↔ (seq𝑀( + , 𝐹)‘𝑧) ∈ (ran + ∪ {∅, (𝐹𝑀)})))
17 fveq2 6842 . . . . . . 7 (𝑦 = (𝑧 + 1) → (seq𝑀( + , 𝐹)‘𝑦) = (seq𝑀( + , 𝐹)‘(𝑧 + 1)))
1817eleq1d 2822 . . . . . 6 (𝑦 = (𝑧 + 1) → ((seq𝑀( + , 𝐹)‘𝑦) ∈ (ran + ∪ {∅, (𝐹𝑀)}) ↔ (seq𝑀( + , 𝐹)‘(𝑧 + 1)) ∈ (ran + ∪ {∅, (𝐹𝑀)})))
19 fveq2 6842 . . . . . . 7 (𝑦 = 𝑥 → (seq𝑀( + , 𝐹)‘𝑦) = (seq𝑀( + , 𝐹)‘𝑥))
2019eleq1d 2822 . . . . . 6 (𝑦 = 𝑥 → ((seq𝑀( + , 𝐹)‘𝑦) ∈ (ran + ∪ {∅, (𝐹𝑀)}) ↔ (seq𝑀( + , 𝐹)‘𝑥) ∈ (ran + ∪ {∅, (𝐹𝑀)})))
21 seq1 13919 . . . . . . 7 (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹𝑀))
22 ssun2 4133 . . . . . . . 8 {∅, (𝐹𝑀)} ⊆ (ran + ∪ {∅, (𝐹𝑀)})
23 fvex 6855 . . . . . . . . 9 (𝐹𝑀) ∈ V
2423prid2 4724 . . . . . . . 8 (𝐹𝑀) ∈ {∅, (𝐹𝑀)}
2522, 24sselii 3941 . . . . . . 7 (𝐹𝑀) ∈ (ran + ∪ {∅, (𝐹𝑀)})
2621, 25eqeltrdi 2846 . . . . . 6 (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) ∈ (ran + ∪ {∅, (𝐹𝑀)}))
27 seqp1 13921 . . . . . . . . 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 7360 . . . . . . . . . 10 ((seq𝑀( + , 𝐹)‘𝑧) + (𝐹‘(𝑧 + 1))) = ( + ‘⟨(seq𝑀( + , 𝐹)‘𝑧), (𝐹‘(𝑧 + 1))⟩)
30 snsspr1 4774 . . . . . . . . . . . 12 {∅} ⊆ {∅, (𝐹𝑀)}
31 unss2 4141 . . . . . . . . . . . 12 ({∅} ⊆ {∅, (𝐹𝑀)} → (ran + ∪ {∅}) ⊆ (ran + ∪ {∅, (𝐹𝑀)}))
3230, 31ax-mp 5 . . . . . . . . . . 11 (ran + ∪ {∅}) ⊆ (ran + ∪ {∅, (𝐹𝑀)})
33 fvrn0 6872 . . . . . . . . . . 11 ( + ‘⟨(seq𝑀( + , 𝐹)‘𝑧), (𝐹‘(𝑧 + 1))⟩) ∈ (ran + ∪ {∅})
3432, 33sselii 3941 . . . . . . . . . 10 ( + ‘⟨(seq𝑀( + , 𝐹)‘𝑧), (𝐹‘(𝑧 + 1))⟩) ∈ (ran + ∪ {∅, (𝐹𝑀)})
3529, 34eqeltri 2834 . . . . . . . . 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 2838 . . . . . . 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 12831 . . . . 5 (𝑥 ∈ (ℤ𝑀) → (seq𝑀( + , 𝐹)‘𝑥) ∈ (ran + ∪ {∅, (𝐹𝑀)}))
4039rgen 3066 . . . 4 𝑥 ∈ (ℤ𝑀)(seq𝑀( + , 𝐹)‘𝑥) ∈ (ran + ∪ {∅, (𝐹𝑀)})
41 fnfvrnss 7068 . . . 4 ((seq𝑀( + , 𝐹) Fn (ℤ𝑀) ∧ ∀𝑥 ∈ (ℤ𝑀)(seq𝑀( + , 𝐹)‘𝑥) ∈ (ran + ∪ {∅, (𝐹𝑀)})) → ran seq𝑀( + , 𝐹) ⊆ (ran + ∪ {∅, (𝐹𝑀)}))
423, 40, 41mp2an 690 . . 3 ran seq𝑀( + , 𝐹) ⊆ (ran + ∪ {∅, (𝐹𝑀)})
4312, 42ssexi 5279 . 2 ran seq𝑀( + , 𝐹) ∈ V
44 funexw 7884 . 2 ((Fun seq𝑀( + , 𝐹) ∧ dom seq𝑀( + , 𝐹) ∈ V ∧ ran seq𝑀( + , 𝐹) ∈ V) → seq𝑀( + , 𝐹) ∈ V)
455, 8, 43, 44mp3an 1461 1 seq𝑀( + , 𝐹) ∈ V
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1541  wcel 2106  wral 3064  Vcvv 3445  cun 3908  wss 3910  c0 4282  {csn 4586  {cpr 4588  cop 4592  dom cdm 5633  ran crn 5634  Fun wfun 6490   Fn wfn 6491  cfv 6496  (class class class)co 7357  1c1 11052   + caddc 11054  cz 12499  cuz 12763  seqcseq 13906
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-seq 13907
This theorem is referenced by:  mulgfval  18874
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