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Theorem nfiseq 9528
Description: Hypothesis builder for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.)
Hypotheses
Ref Expression
nfiseq.1 𝑥𝑀
nfiseq.2 𝑥 +
nfiseq.3 𝑥𝐹
nfiseq.4 𝑥𝑆
Assertion
Ref Expression
nfiseq 𝑥seq𝑀( + , 𝐹, 𝑆)

Proof of Theorem nfiseq
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-iseq 9522 . 2 seq𝑀( + , 𝐹, 𝑆) = ran frec((𝑦 ∈ (ℤ𝑀), 𝑧𝑆 ↦ ⟨(𝑦 + 1), (𝑧 + (𝐹‘(𝑦 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
2 nfcv 2220 . . . . . 6 𝑥
3 nfiseq.1 . . . . . 6 𝑥𝑀
42, 3nffv 5216 . . . . 5 𝑥(ℤ𝑀)
5 nfiseq.4 . . . . 5 𝑥𝑆
6 nfcv 2220 . . . . . 6 𝑥(𝑦 + 1)
7 nfcv 2220 . . . . . . 7 𝑥𝑧
8 nfiseq.2 . . . . . . 7 𝑥 +
9 nfiseq.3 . . . . . . . 8 𝑥𝐹
109, 6nffv 5216 . . . . . . 7 𝑥(𝐹‘(𝑦 + 1))
117, 8, 10nfov 5566 . . . . . 6 𝑥(𝑧 + (𝐹‘(𝑦 + 1)))
126, 11nfop 3594 . . . . 5 𝑥⟨(𝑦 + 1), (𝑧 + (𝐹‘(𝑦 + 1)))⟩
134, 5, 12nfmpt2 5604 . . . 4 𝑥(𝑦 ∈ (ℤ𝑀), 𝑧𝑆 ↦ ⟨(𝑦 + 1), (𝑧 + (𝐹‘(𝑦 + 1)))⟩)
149, 3nffv 5216 . . . . 5 𝑥(𝐹𝑀)
153, 14nfop 3594 . . . 4 𝑥𝑀, (𝐹𝑀)⟩
1613, 15nffrec 6045 . . 3 𝑥frec((𝑦 ∈ (ℤ𝑀), 𝑧𝑆 ↦ ⟨(𝑦 + 1), (𝑧 + (𝐹‘(𝑦 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
1716nfrn 4607 . 2 𝑥ran frec((𝑦 ∈ (ℤ𝑀), 𝑧𝑆 ↦ ⟨(𝑦 + 1), (𝑧 + (𝐹‘(𝑦 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
181, 17nfcxfr 2217 1 𝑥seq𝑀( + , 𝐹, 𝑆)
Colors of variables: wff set class
Syntax hints:  wnfc 2207  cop 3409  ran crn 4372  cfv 4932  (class class class)co 5543  cmpt2 5545  freccfrec 6039  1c1 7044   + caddc 7046  cuz 8700  seqcseq 9521
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-un 2978  df-in 2980  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-mpt 3849  df-xp 4377  df-cnv 4379  df-dm 4381  df-rn 4382  df-res 4383  df-iota 4897  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-recs 5954  df-frec 6040  df-iseq 9522
This theorem is referenced by:  nfsum1  10331  nfsum  10332
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