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Theorem nfseq 10494
Description: Hypothesis builder for the sequence builder operation. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
nfseq.1 𝑥𝑀
nfseq.2 𝑥 +
nfseq.3 𝑥𝐹
Assertion
Ref Expression
nfseq 𝑥seq𝑀( + , 𝐹)

Proof of Theorem nfseq
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-seqfrec 10485 . 2 seq𝑀( + , 𝐹) = ran frec((𝑧 ∈ (ℤ𝑀), 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
2 nfcv 2332 . . . . . 6 𝑥
3 nfseq.1 . . . . . 6 𝑥𝑀
42, 3nffv 5547 . . . . 5 𝑥(ℤ𝑀)
5 nfcv 2332 . . . . 5 𝑥V
6 nfcv 2332 . . . . . 6 𝑥(𝑧 + 1)
7 nfcv 2332 . . . . . . 7 𝑥𝑤
8 nfseq.2 . . . . . . 7 𝑥 +
9 nfseq.3 . . . . . . . 8 𝑥𝐹
109, 6nffv 5547 . . . . . . 7 𝑥(𝐹‘(𝑧 + 1))
117, 8, 10nfov 5930 . . . . . 6 𝑥(𝑤 + (𝐹‘(𝑧 + 1)))
126, 11nfop 3812 . . . . 5 𝑥⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩
134, 5, 12nfmpo 5969 . . . 4 𝑥(𝑧 ∈ (ℤ𝑀), 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩)
149, 3nffv 5547 . . . . 5 𝑥(𝐹𝑀)
153, 14nfop 3812 . . . 4 𝑥𝑀, (𝐹𝑀)⟩
1613, 15nffrec 6425 . . 3 𝑥frec((𝑧 ∈ (ℤ𝑀), 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
1716nfrn 4893 . 2 𝑥ran frec((𝑧 ∈ (ℤ𝑀), 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
181, 17nfcxfr 2329 1 𝑥seq𝑀( + , 𝐹)
Colors of variables: wff set class
Syntax hints:  wnfc 2319  Vcvv 2752  cop 3613  ran crn 4648  cfv 5238  (class class class)co 5900  cmpo 5902  freccfrec 6419  1c1 7847   + caddc 7849  cuz 9563  seqcseq 10484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-un 3148  df-in 3150  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-br 4022  df-opab 4083  df-mpt 4084  df-xp 4653  df-cnv 4655  df-dm 4657  df-rn 4658  df-res 4659  df-iota 5199  df-fv 5246  df-ov 5903  df-oprab 5904  df-mpo 5905  df-recs 6334  df-frec 6420  df-seqfrec 10485
This theorem is referenced by:  seq3f1olemstep  10540  seq3f1olemp  10541  nfsum1  11405  nfsum  11406  nfcprod1  11603  nfcprod  11604
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