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Theorem nfseq 13005
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-seq 12996 . 2 seq𝑀( + , 𝐹) = (rec((𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) “ ω)
2 nfcv 2902 . . . . 5 𝑥V
3 nfcv 2902 . . . . . 6 𝑥(𝑧 + 1)
4 nfcv 2902 . . . . . . 7 𝑥𝑤
5 nfseq.2 . . . . . . 7 𝑥 +
6 nfseq.3 . . . . . . . 8 𝑥𝐹
76, 3nffv 6359 . . . . . . 7 𝑥(𝐹‘(𝑧 + 1))
84, 5, 7nfov 6839 . . . . . 6 𝑥(𝑤 + (𝐹‘(𝑧 + 1)))
93, 8nfop 4569 . . . . 5 𝑥⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩
102, 2, 9nfmpt2 6889 . . . 4 𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩)
11 nfseq.1 . . . . 5 𝑥𝑀
126, 11nffv 6359 . . . . 5 𝑥(𝐹𝑀)
1311, 12nfop 4569 . . . 4 𝑥𝑀, (𝐹𝑀)⟩
1410, 13nfrdg 7679 . . 3 𝑥rec((𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
15 nfcv 2902 . . 3 𝑥ω
1614, 15nfima 5632 . 2 𝑥(rec((𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) “ ω)
171, 16nfcxfr 2900 1 𝑥seq𝑀( + , 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wnfc 2889  Vcvv 3340  cop 4327  cima 5269  cfv 6049  (class class class)co 6813  cmpt2 6815  ωcom 7230  reccrdg 7674  1c1 10129   + caddc 10131  seqcseq 12995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-xp 5272  df-cnv 5274  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-iota 6012  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-seq 12996
This theorem is referenced by:  seqof2  13053  nfsum1  14619  nfsum  14620  nfcprod1  14839  nfcprod  14840  lgamgulm2  24961  binomcxplemdvbinom  39054  binomcxplemdvsum  39056  binomcxplemnotnn0  39057  fmuldfeqlem1  40317  fmuldfeq  40318  sumnnodd  40365  stoweidlem51  40771
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