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Theorem nfiseq 9058
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 9052 . 2 seq𝑀( + , 𝐹, 𝑆) = ran frec((𝑦 ∈ (ℤ𝑀), 𝑧𝑆 ↦ ⟨(𝑦 + 1), (𝑧 + (𝐹‘(𝑦 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
2 nfcv 2178 . . . . . 6 𝑥
3 nfiseq.1 . . . . . 6 𝑥𝑀
42, 3nffv 5146 . . . . 5 𝑥(ℤ𝑀)
5 nfiseq.4 . . . . 5 𝑥𝑆
6 nfcv 2178 . . . . . 6 𝑥(𝑦 + 1)
7 nfcv 2178 . . . . . . 7 𝑥𝑧
8 nfiseq.2 . . . . . . 7 𝑥 +
9 nfiseq.3 . . . . . . . 8 𝑥𝐹
109, 6nffv 5146 . . . . . . 7 𝑥(𝐹‘(𝑦 + 1))
117, 8, 10nfov 5496 . . . . . 6 𝑥(𝑧 + (𝐹‘(𝑦 + 1)))
126, 11nfop 3561 . . . . 5 𝑥⟨(𝑦 + 1), (𝑧 + (𝐹‘(𝑦 + 1)))⟩
134, 5, 12nfmpt2 5534 . . . 4 𝑥(𝑦 ∈ (ℤ𝑀), 𝑧𝑆 ↦ ⟨(𝑦 + 1), (𝑧 + (𝐹‘(𝑦 + 1)))⟩)
149, 3nffv 5146 . . . . 5 𝑥(𝐹𝑀)
153, 14nfop 3561 . . . 4 𝑥𝑀, (𝐹𝑀)⟩
1613, 15nffrec 5943 . . 3 𝑥frec((𝑦 ∈ (ℤ𝑀), 𝑧𝑆 ↦ ⟨(𝑦 + 1), (𝑧 + (𝐹‘(𝑦 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
1716nfrn 4540 . 2 𝑥ran frec((𝑦 ∈ (ℤ𝑀), 𝑧𝑆 ↦ ⟨(𝑦 + 1), (𝑧 + (𝐹‘(𝑦 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
181, 17nfcxfr 2175 1 𝑥seq𝑀( + , 𝐹, 𝑆)
Colors of variables: wff set class
Syntax hints:  wnfc 2165  cop 3375  ran crn 4307  cfv 4863  (class class class)co 5473  cmpt2 5475  freccfrec 5938  1c1 6833   + caddc 6835  cuz 8407  seqcseq 9051
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rex 2309  df-rab 2312  df-v 2556  df-un 2919  df-in 2921  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3577  df-br 3761  df-opab 3815  df-mpt 3816  df-xp 4312  df-cnv 4314  df-dm 4316  df-rn 4317  df-res 4318  df-iota 4828  df-fv 4871  df-ov 5476  df-oprab 5477  df-mpt2 5478  df-recs 5881  df-frec 5939  df-iseq 9052
This theorem is referenced by:  nfsum1  9714  nfsum  9715
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